Image coding method based on secondary transform, and device therefor

ABSTRACT

An image decoding method according to the present document comprises the steps of: deriving transform coefficients through inverse quantization on the basis of quantized transform coefficients for a target block; deriving modified transform coefficients on the basis of an inverse reduced secondary transform (RST) for the transform coefficients; and generating, on the basis of an inverse primary transform for the modified transform coefficients, a restoration picture based on residual samples for the target block, wherein the modified transform coefficients derived according to the inverse RST are two-dimensionally arranged according to the order of a row priority direction or a column priority direction according to an intra prediction mode to be applied to the target block.

CROSS-REFERENCE TO RELATED APPLICATIONS

Pursuant to 35 U.S.C. § 119(e), this application is a continuation ofInternational Application PCT/KR2019/017892, with an internationalfiling date of Dec. 17, 2019, which claims the benefit of U.S.Provisional Patent Application No. 62/781,594, filed on Dec. 18, 2018,the contents of which are hereby incorporated by reference herein intheir entirety.

TECHNICAL FIELD

The present disclosure relates generally to an image coding technologyand, more particularly, to an image coding method based on a transformin an image coding system and an apparatus therefor.

RELATED ART

Nowadays, the demand for high-resolution and high-quality images/videossuch as 4K, 8K or more ultra high definition (UHD) images/videos hasbeen increasing in various fields. As the image/video data becomeshigher resolution and higher quality, the transmitted information amountor bit amount increases as compared to the conventional image data.Therefore, when image data is transmitted using a medium such as aconventional wired/wireless broadband line or image/video data is storedusing an existing storage medium, the transmission cost and the storagecost thereof are increased.

Further, nowadays, the interest and demand for immersive media such asvirtual reality (VR), artificial reality (AR) content or hologram, orthe like is increasing, and broadcasting for images/videos having imagefeatures different from those of real images, such as a game image isincreasing.

Accordingly, there is a need for a highly efficient image/videocompression technique for effectively compressing and transmitting orstoring, and reproducing information of high resolution and high qualityimages/videos having various features as described above.

SUMMARY

A technical aspect of the present disclosure is to provide a method andan apparatus for increasing image coding efficiency.

Another technical aspect of the present disclosure is to provide amethod and an apparatus for increasing transform efficiency.

Still another technical aspect of the present disclosure is to providean image coding method and an image coding apparatus which are based ona reduced secondary transform (RST).

Yet another technical aspect of the present disclosure is to provide amethod and an apparatus for increasing the efficiency of a secondarytransform by changing the array of transform coefficients according toan intra prediction mode.

Still another technical aspect of the present disclosure is to providean image coding method and an image coding apparatus which are based ona transform set for increasing coding efficiency.

According to an embodiment of the present disclosure, there is providedan image decoding method performed by a decoding apparatus. The methodmay include: deriving transform coefficients through dequantizationbased on quantized transform coefficients for a target block; derivingmodified transform coefficients based on an inverse reduced secondarytransform (RST) using a preset transform kernel matrix for the transformcoefficients; deriving residual samples for the target block based on aninverse primary transform for the modified transform coefficients; andgenerating a reconstructed picture based on the residual samples for thetarget block, wherein the modified transform coefficients in atwo-dimensional array may be derived, by applying the inverse RST, fromthe transform coefficients in a one-dimensional (1D) array according toone of a row-first direction or a column-first direction according to anintra prediction mode applied to the target block.

According to another embodiment of the present disclosure, there isprovided a decoding apparatus for performing image decoding. Thedecoding apparatus may include: an entropy decoder to derive quantizedtransform coefficients for a target block and information on predictionfrom a bitstream; a predictor to generate a prediction sample for thetarget block based on the information on prediction; a dequantizer toderive transform coefficients through dequantization based on thequantized transform coefficients for the target block; an inversetransformer to include an inverse reduced secondary transformer (RST)that derives modified transform coefficients based on inverse RST of thetransform coefficients and an inverse primary transformer that derivesresidual samples for the target block based on first inverse transformof the modified transform coefficients; and an adder to generatereconstructed samples based on the residual samples and the predictionsamples, wherein the modified transform coefficients in atwo-dimensional array may be derived, by applying the inverse RST, fromthe transform coefficients in a one-dimensional (1D) array according toone of a row-first direction or a column-first direction according to anintra prediction mode applied to the target block.

According to still another embodiment of the present disclosure, thereis provided an image encoding method performed by an encoding apparatus.The method may include: deriving prediction samples based on an intraprediction mode applied to a target block; deriving residual samples forthe target block based on the prediction samples; deriving transformcoefficients for the target block based on a primary transform for theresidual samples; deriving modified transform coefficients based on areduced secondary transform (RST) for the transform coefficients; andderiving quantized transform coefficients by performing quantizationbased on the modified transform coefficients, wherein the deriving ofthe modified transform coefficients one-dimensionally may includearranging the transform coefficients according to one of a row-firstdirection or a column-first direction according to the intra predictionmode applied to the target block and applying the RST to the transformcoefficients.

According to yet another embodiment of the present disclosure, there maybe provided a digital storage medium that stores image data includingencoded image information generated according to an image encodingmethod performed by an encoding apparatus.

According to still another embodiment of the present disclosure, theremay be provided a digital storage medium that stores image dataincluding encoded image information or a bitstream to cause a decodingapparatus to perform the image decoding method.

According to the present disclosure, it is possible to increase overallimage/video compression efficiency.

According to the present disclosure, it is possible to increase theefficiency of a secondary transform by changing the array of transformcoefficients according to an intra prediction mode.

According to the present disclosure, it is possible to increase imagecoding efficiency by performing image coding based on a transform set.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically illustrates an example of a video/image codingsystem to which the present disclosure is applicable.

FIG. 2 is a diagram schematically illustrating a configuration of avideo/image encoding apparatus to which the present disclosure isapplicable.

FIG. 3 is a diagram schematically illustrating a configuration of avideo/image decoding apparatus to which the present disclosure isapplicable.

FIG. 4 schematically illustrates a multiple transform techniqueaccording to an embodiment of the present disclosure.

FIG. 5 illustrates directional intra modes of 65 prediction directions.

FIG. 6 is a diagram illustrating an RST according to an embodiment ofthe present disclosure.

FIG. 7 is a diagram illustrating a transform coefficient scanning orderaccording to an embodiment of the present disclosure.

FIG. 8 is a flowchart illustrating an inverse RST process according toan embodiment of the present disclosure.

FIG. 9 is a flowchart illustrating an operation of a video decodingapparatus according to an embodiment of the present disclosure.

FIG. 10 is a control flowchart illustrating an inverse RST according toan embodiment of the present disclosure.

FIG. 11 is a flowchart illustrating an operation of a video encodingapparatus according to an embodiment of the present disclosure.

FIG. 12 is a control flowchart illustrating an RST according to anembodiment of the present disclosure.

FIG. 13 illustrates the structure of a content streaming system to whichthe present disclosure is applied.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

While the present disclosure may be susceptible to various modificationsand include various embodiments, specific embodiments thereof have beenshown in the drawings by way of example and will now be described indetail. However, this is not intended to limit the present disclosure tothe specific embodiments disclosed herein. The terminology used hereinis for the purpose of describing specific embodiments only, and is notintended to limit technical idea of the present disclosure. The singularforms may include the plural forms unless the context clearly indicatesotherwise. The terms such as “include” and “have” are intended toindicate that features, numbers, steps, operations, elements,components, or combinations thereof used in the following descriptionexist, and thus should not be understood as that the possibility ofexistence or addition of one or more different features, numbers, steps,operations, elements, components, or combinations thereof is excluded inadvance.

Meanwhile, each component on the drawings described herein isillustrated independently for convenience of description as tocharacteristic functions different from each other, and however, it isnot meant that each component is realized by a separate hardware orsoftware. For example, any two or more of these components may becombined to form a single component, and any single component may bedivided into plural components. The embodiments in which components arecombined and/or divided will belong to the scope of the patent right ofthe present disclosure as long as they do not depart from the essence ofthe present disclosure.

Hereinafter, preferred embodiments of the present disclosure will beexplained in more detail while referring to the attached drawings. Inaddition, the same reference signs are used for the same components onthe drawings, and repeated descriptions for the same components will beomitted.

This document relates to video/image coding. For example, themethod/example disclosed in this document may relate to a VVC (VersatileVideo Coding) standard (ITU-T Rec. H.266), a next-generation video/imagecoding standard after VVC, or other video coding related standards(e.g., HEVC (High Efficiency Video Coding) standard (ITU-T Rec. H.265),EVC (essential video coding) standard, AVS2 standard, etc.).

In this document, a variety of embodiments relating to video/imagecoding may be provided, and, unless specified to the contrary, theembodiments may be combined to each other and be performed.

In this document, a video may mean a set of a series of images overtime. Generally a picture means a unit representing an image at aspecific time zone, and a slice/tile is a unit constituting a part ofthe picture. The slice/tile may include one or more coding tree units(CTUs). One picture may be constituted by one or more slices/tiles. Onepicture may be constituted by one or more tile groups. One tile groupmay include one or more tiles.

A pixel or a pel may mean a smallest unit constituting one picture (orimage). Also, ‘sample’ may be used as a term corresponding to a pixel. Asample may generally represent a pixel or a value of a pixel, and mayrepresent only a pixel/pixel value of a luma component or only apixel/pixel value of a chroma component. Alternatively, the sample mayrefer to a pixel value in the spatial domain, or when this pixel valueis converted to the frequency domain, it may refer to a transformcoefficient in the frequency domain.

A unit may represent the basic unit of image processing. The unit mayinclude at least one of a specific region and information related to theregion. One unit may include one luma block and two chroma (e.g., cb,cr) blocks. The unit and a term such as a block, an area, or the likemay be used in place of each other according to circumstances. In ageneral case, an M×N block may include a set (or an array) of samples(or sample arrays) or transform coefficients consisting of M columns andN rows.

In this document, the term “/” and “,” should be interpreted to indicate“and/or.” For instance, the expression “A/B” may mean “A and/or B.”Further, “A, B” may mean “A and/or B.” Further, “A/B/C” may mean “atleast one of A, B, and/or C.” Also, “A/B/C” may mean “at least one of A,B, and/or C.”

Further, in the document, the term “or” should be interpreted toindicate “and/or.” For instance, the expression “A or B” may include 1)only A, 2) only B, and/or 3) both A and B. In other words, the term “or”in this document should be interpreted to indicate “additionally oralternatively.”

FIG. 1 schematically illustrates an example of a video/image codingsystem to which the present disclosure is applicable.

Referring to FIG. 1, the video/image coding system may include a firstdevice (source device) and a second device (receive device). The sourcedevice may deliver encoded video/image information or data in the formof a file or streaming to the receive device via a digital storagemedium or network.

The source device may include a video source, an encoding apparatus, anda transmitter. The receive device may include a receiver, a decodingapparatus, and a renderer. The encoding apparatus may be called avideo/image encoding apparatus, and the decoding apparatus may be calleda video/image decoding apparatus. The transmitter may be included in theencoding apparatus. The receiver may be included in the decodingapparatus. The renderer may include a display, and the display may beconfigured as a separate device or an external component.

The video source may obtain a video/image through a process ofcapturing, synthesizing, or generating a video/image. The video sourcemay include a video/image capture device and/or a video/image generatingdevice. The video/image capture device may include, for example, one ormore cameras, video/image archives including previously capturedvideo/images, or the like. The video/image generating device mayinclude, for example, a computer, a tablet and a smartphone, and may(electronically) generate a video/image. For example, a virtualvideo/image may be generated through a computer or the like. In thiscase, the video/image capturing process may be replaced by a process ofgenerating related data.

The encoding apparatus may encode an input video/image. The encodingapparatus may perform a series of procedures such as prediction,transform, and quantization for compression and coding efficiency. Theencoded data (encoded video/image information) may be output in the formof a bitstream.

The transmitter may transmit the encoded video/image information or dataoutput in the form of a bitstream to the receiver of the receive devicethrough a digital storage medium or a network in the form of a file orstreaming. The digital storage medium may include various storagemediums such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, and the like. Thetransmitter may include an element for generating a media file through apredetermined file format, and may include an element for transmissionthrough a broadcast/communication network. The receiver mayreceive/extract the bitstream and transmit the received/extractedbitstream to the decoding apparatus.

The decoding apparatus may decode a video/image by performing a seriesof procedures such as dequantization, inverse transform, prediction, andthe like corresponding to the operation of the encoding apparatus.

The renderer may render the decoded video/image. The renderedvideo/image may be displayed through the display.

FIG. 2 is a diagram schematically illustrating a configuration of avideo/image encoding apparatus to which the present disclosure isapplicable. Hereinafter, what is referred to as the video encodingapparatus may include an image encoding apparatus.

Referring to FIG. 2, the encoding apparatus 200 may include an imagepartitioner 210, a predictor 220, a residual processor 230, an entropyencoder 240, an adder 250, a filter 260, and a memory 270. The predictor220 may include an inter predictor 221 and an intra predictor 222. Theresidual processor 230 may include a transformer 232, a quantizer 233, adequantizer 234, an inverse transformer 235. The residual processor 230may further include a subtractor 231. The adder 250 may be called areconstructor or reconstructed block generator. The image partitioner210, the predictor 220, the residual processor 230, the entropy encoder240, the adder 250, and the filter 260, which have been described above,may be constituted by one or more hardware components (e.g., encoderchipsets or processors) according to an embodiment. Further, the memory270 may include a decoded picture buffer (DPB), and may be constitutedby a digital storage medium. The hardware component may further includethe memory 270 as an internal/external component.

The image partitioner 210 may partition an input image (or a picture ora frame) input to the encoding apparatus 200 into one or more processingunits. As one example, the processing unit may be called a coding unit(CU). In this case, starting with a coding tree unit (CTU) or thelargest coding unit (LCU), the coding unit may be recursivelypartitioned according to the Quad-tree binary-tree ternary-tree (QTBTTT)structure. For example, one coding unit may be divided into a pluralityof coding units of a deeper depth based on the quad-tree structure, thebinary-tree structure, and/or the ternary structure. In this case, forexample, the quad-tree structure may be applied first and thebinary-tree structure and/or the ternary structure may be applied later.Alternatively, the binary-tree structure may be applied first. Thecoding procedure according to the present disclosure may be performedbased on the final coding unit which is not further partitioned. In thiscase, the maximum coding unit may be used directly as a final codingunit based on coding efficiency according to the image characteristic.Alternatively, the coding unit may be recursively partitioned intocoding units of a further deeper depth as needed, so that the codingunit of an optimal size may be used as a final coding unit. Here, thecoding procedure may include procedures such as prediction, transform,and reconstruction, which will be described later. As another example,the processing unit may further include a prediction unit (PU) or atransform unit (TU). In this case, the prediction unit and the transformunit may be split or partitioned from the above-described final codingunit. The prediction unit may be a unit of sample prediction, and thetransform unit may be a unit for deriving a transform coefficient and/ora unit for deriving a residual signal from a transform coefficient.

The unit and a term such as a block, an area, or the like may be used inplace of each other according to circumstances. In a general case, anM×N block may represent a set of samples or transform coefficientsconsisting of M columns and N rows. The sample may generally represent apixel or a value of a pixel, and may represent only a pixel/pixel valueof a luma component, or only a pixel/pixel value of a chroma component.The sample may be used as a term corresponding to a pixel or a pel ofone picture (or image).

The subtractor 231 subtracts a prediction signal (predicted block,prediction sample array) output from the inter predictor 221 or theintra predictor 222 from an input image signal (original block, originalsample array) to generate a residual signal (residual block, residualsample array), and the generated residual signal is transmitted to thetransformer 232. In this case, as shown, a unit which subtracts theprediction signal (predicted block, prediction sample array) from theinput image signal (original block, original sample array) in theencoder 200 may be called the subtractor 231. The predictor may performprediction on a processing target block (hereinafter, referred to as‘current block’), and may generate a predicted block includingprediction samples for the current block. The predictor may determinewhether intra prediction or inter prediction is applied on a currentblock or CU basis. As discussed later in the description of eachprediction mode, the predictor may generate various information relatingto prediction, such as prediction mode information, and transmit thegenerated information to the entropy encoder 240. The information on theprediction may be encoded in the entropy encoder 240 and output in theform of a bitstream.

The intra predictor 222 may predict the current block by referring tosamples in the current picture. The referred samples may be located inthe neighbor of or apart from the current block according to theprediction mode. In the intra prediction, prediction modes may include aplurality of non-directional modes and a plurality of directional modes.The non-directional modes may include, for example, a DC mode and aplanar mode. The directional mode may include, for example, 33directional prediction modes or 65 directional prediction modesaccording to the degree of detail of the prediction direction. However,this is merely an example, and more or less directional prediction modesmay be used depending on a setting. The intra predictor 222 maydetermine the prediction mode applied to the current block by using theprediction mode applied to the neighboring block.

The inter predictor 221 may derive a predicted block for the currentblock based on a reference block (reference sample array) specified by amotion vector on a reference picture. At this time, in order to reducethe amount of motion information transmitted in the inter predictionmode, the motion information may be predicted on a block, subblock, orsample basis based on correlation of motion information between theneighboring block and the current block. The motion information mayinclude a motion vector and a reference picture index. The motioninformation may further include inter prediction direction (L0prediction, L1 prediction, Bi prediction, etc.) information. In the caseof inter prediction, the neighboring block may include a spatialneighboring block existing in the current picture and a temporalneighboring block existing in the reference picture. The referencepicture including the reference block and the reference pictureincluding the temporal neighboring block may be same to each other ordifferent from each other. The temporal neighboring block may be calleda collocated reference block, a collocated CU (colCU), and the like, andthe reference picture including the temporal neighboring block may becalled a collocated picture (colPic). For example, the inter predictor221 may configure a motion information candidate list based onneighboring blocks and generate information indicating which candidateis used to derive a motion vector and/or a reference picture index ofthe current block. Inter prediction may be performed based on variousprediction modes. For example, in the case of a skip mode and a mergemode, the inter predictor 221 may use motion information of theneighboring block as motion information of the current block. In theskip mode, unlike the merge mode, the residual signal may not betransmitted. In the case of the motion information prediction (motionvector prediction, MVP) mode, the motion vector of the neighboring blockmay be used as a motion vector predictor and the motion vector of thecurrent block may be indicated by signaling a motion vector difference.

The predictor 220 may generate a prediction signal based on variousprediction methods. For example, the predictor may apply intraprediction or inter prediction for prediction on one block, and, aswell, may apply intra prediction and inter prediction at the same time.This may be called combined inter and intra prediction (CIIP). Further,the predictor may be based on an intra block copy (IBC) prediction mode,or a palette mode in order to perform prediction on a block. The IBCprediction mode or palette mode may be used for content image/videocoding of a game or the like, such as screen content coding (SCC).Although the IBC basically performs prediction in a current block, itcan be performed similarly to inter prediction in that it derives areference block in a current block. That is, the IBC may use at leastone of inter prediction techniques described in the present disclosure.

The prediction signal generated through the inter predictor 221 and/orthe intra predictor 222 may be used to generate a reconstructed signalor to generate a residual signal. The transformer 232 may generatetransform coefficients by applying a transform technique to the residualsignal. For example, the transform technique may include at least one ofa discrete cosine transform (DCT), a discrete sine transform (DST), aKarhunen-Loève transform (KLT), a graph-based transform (GBT), or aconditionally non-linear transform (CNT). Here, the GBT means transformobtained from a graph when relationship information between pixels isrepresented by the graph. The CNT refers to transform obtained based ona prediction signal generated using all previously reconstructed pixels.In addition, the transform process may be applied to square pixel blockshaving the same size or may be applied to blocks having a variable sizerather than the square one.

The quantizer 233 may quantize the transform coefficients and transmitthem to the entropy encoder 240, and the entropy encoder 240 may encodethe quantized signal (information on the quantized transformcoefficients) and output the encoded signal in a bitstream. Theinformation on the quantized transform coefficients may be referred toas residual information. The quantizer 233 may rearrange block typequantized transform coefficients into a one-dimensional vector formbased on a coefficient scan order, and generate information on thequantized transform coefficients based on the quantized transformcoefficients of the one-dimensional vector form. The entropy encoder 240may perform various encoding methods such as, for example, exponentialGolomb, context-adaptive variable length coding (CAVLC),context-adaptive binary arithmetic coding (CABAC), and the like. Theentropy encoder 240 may encode information necessary for video/imagereconstruction other than quantized transform coefficients (e.g. valuesof syntax elements, etc.) together or separately. Encoded information(e.g., encoded video/image information) may be transmitted or stored ona unit basis of a network abstraction layer (NAL) in the form of abitstream. The video/image information may further include informationon various parameter sets such as an adaptation parameter set (APS), apicture parameter set (PPS), a sequence parameter set (SPS), a videoparameter set (VPS) or the like. Further, the video/image informationmay further include general constraint information. In the presentdisclosure, information and/or syntax elements which aretransmitted/signaled to the decoding apparatus from the encodingapparatus may be included in video/image information. The video/imageinformation may be encoded through the above-described encodingprocedure and included in the bitstream. The bitstream may betransmitted through a network, or stored in a digital storage medium.Here, the network may include a broadcast network, a communicationnetwork and/or the like, and the digital storage medium may includevarious storage media such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, andthe like. A transmitter (not shown) which transmits a signal output fromthe entropy encoder 240 and/or a storage (not shown) which stores it maybe configured as an internal/external element of the encoding apparatus200, or the transmitter may be included in the entropy encoder 240.

Quantized transform coefficients output from the quantizer 233 may beused to generate a prediction signal. For example, by applyingdequantization and inverse transform to quantized transform coefficientsthrough the dequantizer 234 and the inverse transformer 235, theresidual signal (residual block or residual samples) may bereconstructed. The adder 155 adds the reconstructed residual signal to aprediction signal output from the inter predictor 221 or the intrapredictor 222, so that a reconstructed signal (reconstructed picture,reconstructed block, reconstructed sample array) may be generated. Whenthere is no residual for a processing target block as in a case wherethe skip mode is applied, the predicted block may be used as areconstructed block. The adder 250 may be called a reconstructor or areconstructed block generator. The generated reconstructed signal may beused for intra prediction of a next processing target block in thecurrent block, and as described later, may be used for inter predictionof a next picture through filtering.

Meanwhile, in the picture encoding and/or reconstructing process, lumamapping with chroma scaling (LMCS) may be applied.

The filter 260 may improve subjective/objective video quality byapplying the filtering to the reconstructed signal. For example, thefilter 260 may generate a modified reconstructed picture by applyingvarious filtering methods to the reconstructed picture, and may storethe modified reconstructed picture in the memory 270, specifically inthe DPB of the memory 270. The various filtering methods may include,for example, deblocking filtering, sample adaptive offset, an adaptiveloop filter, a bilateral filter or the like. As discussed later in thedescription of each filtering method, the filter 260 may generatevarious information relating to filtering, and transmit the generatedinformation to the entropy encoder 240. The information on the filteringmay be encoded in the entropy encoder 240 and output in the form of abitstream.

The modified reconstructed picture which has been transmitted to thememory 270 may be used as a reference picture in the inter predictor221. Through this, the encoding apparatus can avoid prediction mismatchin the encoding apparatus 100 and a decoding apparatus when the interprediction is applied, and can also improve coding efficiency.

The memory 270 DPB may store the modified reconstructed picture in orderto use it as a reference picture in the inter predictor 221. The memory270 may store motion information of a block in the current picture, fromwhich motion information has been derived (or encoded) and/or motioninformation of blocks in an already reconstructed picture. The storedmotion information may be transmitted to the inter predictor 221 to beutilized as motion information of a neighboring block or motioninformation of a temporal neighboring block. The memory 270 may storereconstructed samples of reconstructed blocks in the current picture,and transmit them to the intra predictor 222.

FIG. 3 is a diagram schematically illustrating a configuration of avideo/image decoding apparatus to which the present disclosure isapplicable.

Referring to FIG. 3, the video decoding apparatus 300 may include anentropy decoder 310, a residual processor 320, a predictor 330, an adder340, a filter 350 and a memory 360. The predictor 330 may include aninter predictor 331 and an intra predictor 332. The residual processor320 may include a dequantizer 321 and an inverse transformer 321. Theentropy decoder 310, the residual processor 320, the predictor 330, theadder 340, and the filter 350, which have been described above, may beconstituted by one or more hardware components (e.g., decoder chipsetsor processors) according to an embodiment. Further, the memory 360 mayinclude a decoded picture buffer (DPB), and may be constituted by adigital storage medium. The hardware component may further include thememory 360 as an internal/external component.

When a bitstream including video/image information is input, thedecoding apparatus 300 may reconstruct an image correspondingly to aprocess by which video/image information has been processed in theencoding apparatus of FIG. 2. For example, the decoding apparatus 300may derive units/blocks based on information relating to block partitionobtained from the bitstream. The decoding apparatus 300 may performdecoding by using a processing unit applied in the encoding apparatus.Therefore, the processing unit of decoding may be, for example, a codingunit, which may be partitioned along the quad-tree structure, thebinary-tree structure, and/or the ternary-tree structure from a codingtree unit or a largest coding unit. One or more transform units may bederived from the coding unit. And, the reconstructed image signaldecoded and output through the decoding apparatus 300 may be reproducedthrough a reproducer.

The decoding apparatus 300 may receive a signal output from the encodingapparatus of FIG. 2 in the form of a bitstream, and the received signalmay be decoded through the entropy decoder 310. For example, the entropydecoder 310 may parse the bitstream to derive information (e.g.,video/image information) required for image reconstruction (or picturereconstruction). The video/image information may further includeinformation on various parameter sets such as an adaptation parameterset (APS), a picture parameter set (PPS), a sequence parameter set(SPS), a video parameter set (VPS) or the like. Further, the video/imageinformation may further include general constraint information. Thedecoding apparatus may decode a picture further based on information onthe parameter set and/or the general constraint information. In thepresent disclosure, signaled/received information and/or syntaxelements, which will be described later, may be decoded through thedecoding procedure and be obtained from the bitstream. For example, theentropy decoder 310 may decode information in the bitstream based on acoding method such as exponential Golomb encoding, CAVLC, CABAC, or thelike, and may output a value of a syntax element necessary for imagereconstruction and quantized values of a transform coefficient regardinga residual. More specifically, a CABAC entropy decoding method mayreceive a bin corresponding to each syntax element in a bitstream,determine a context model using decoding target syntax elementinformation and decoding information of neighboring and decoding targetblocks, or information of symbol/bin decoded in a previous step, predictbin generation probability according to the determined context model andperform arithmetic decoding of the bin to generate a symbolcorresponding to each syntax element value. Here, the CABAC entropydecoding method may update the context model using information of asymbol/bin decoded for a context model of the next symbol/bin afterdetermination of the context model. Information on prediction amonginformation decoded in the entropy decoder 310 may be provided to thepredictor (inter predictor 332 and intra predictor 331), and residualvalues, that is, quantized transform coefficients, on which entropydecoding has been performed in the entropy decoder 310, and associatedparameter information may be input to the residual processor 320. Theresidual processor 320 may derive a residual signal (residual block,residual samples, residual sample array). Further, information onfiltering among information decoded in the entropy decoder 310 may beprovided to the filter 350. Meanwhile, a receiver (not shown) whichreceives a signal output from the encoding apparatus may furtherconstitute the decoding apparatus 300 as an internal/external element,and the receiver may be a component of the entropy decoder 310.Meanwhile, the decoding apparatus according to the present disclosuremay be called a video/image/picture coding apparatus, and the decodingapparatus may be classified into an information decoder(video/image/picture information decoder) and a sample decoder(video/image/picture sample decoder). The information decoder mayinclude the entropy decoder 310, and the sample decoder may include atleast one of the dequantizer 321, the inverse transformer 322, the adder340, the filter 350, the memory 360, the inter predictor 332, and theintra predictor 331.

The dequantizer 321 may output transform coefficients by dequantizingthe quantized transform coefficients. The dequantizer 321 may rearrangethe quantized transform coefficients in the form of a two-dimensionalblock. In this case, the rearrangement may perform rearrangement basedon an order of coefficient scanning which has been performed in theencoding apparatus. The dequantizer 321 may perform dequantization onthe quantized transform coefficients using quantization parameter (e.g.,quantization step size information), and obtain transform coefficients.

The deqauntizer 322 obtains a residual signal (residual block, residualsample array) by inverse transforming transform coefficients.

The predictor may perform prediction on the current block, and generatea predicted block including prediction samples for the current block.The predictor may determine whether intra prediction or inter predictionis applied to the current block based on the information on predictionoutput from the entropy decoder 310, and specifically may determine anintra/inter prediction mode.

The predictor may generate a prediction signal based on variousprediction methods. For example, the predictor may apply intraprediction or inter prediction for prediction on one block, and, aswell, may apply intra prediction and inter prediction at the same time.This may be called combined inter and intra prediction (CIIP). Inaddition, the predictor may perform intra block copy (IBC) forprediction on a block. The intra block copy may be used for contentimage/video coding of a game or the like, such as screen content coding(SCC). Although the IBC basically performs prediction in a currentblock, it can be performed similarly to inter prediction in that itderives a reference block in a current block. That is, the IBC may useat least one of inter prediction techniques described in the presentdisclosure.

The intra predictor 331 may predict the current block by referring tothe samples in the current picture. The referred samples may be locatedin the neighbor of or apart from the current block according to theprediction mode. In the intra prediction, prediction modes may include aplurality of non-directional modes and a plurality of directional modes.The intra predictor 331 may determine the prediction mode applied to thecurrent block by using the prediction mode applied to the neighboringblock.

The inter predictor 332 may derive a predicted block for the currentblock based on a reference block (reference sample array) specified by amotion vector on a reference picture. At this time, in order to reducethe amount of motion information transmitted in the inter predictionmode, the motion information may be predicted on a block, subblock, orsample basis based on correlation of motion information between theneighboring block and the current block. The motion information mayinclude a motion vector and a reference picture index. The motioninformation may further include inter prediction direction (L0prediction, L1 prediction, Bi prediction, etc.) information. In the caseof inter prediction, the neighboring block may include a spatialneighboring block existing in the current picture and a temporalneighboring block existing in the reference picture. For example, theinter predictor 332 may configure a motion information candidate listbased on neighboring blocks, and derive a motion vector and/or areference picture index of the current block based on received candidateselection information. Inter prediction may be performed based onvarious prediction modes, and the information on prediction may includeinformation indicating a mode of inter prediction for the current block.

The adder 340 may generate a reconstructed signal (reconstructedpicture, reconstructed block, reconstructed sample array) by adding theobtained residual signal to the prediction signal (predicted block,prediction sample array) output from the predictor 330. When there is noresidual for a processing target block as in a case where the skip modeis applied, the predicted block may be used as a reconstructed block.

The adder 340 may be called a reconstructor or a reconstructed blockgenerator. The generated reconstructed signal may be used for intraprediction of a next processing target block in the current block, andas described later, may be output through filtering or be used for interprediction of a next picture.

Meanwhile, in the picture decoding process, luma mapping with chromascaling (LMCS) may be applied.

The filter 350 may improve subjective/objective video quality byapplying the filtering to the reconstructed signal. For example, thefilter 350 may generate a modified reconstructed picture by applyingvarious filtering methods to the reconstructed picture, and may transmitthe modified reconstructed picture in the memory 360, specifically inthe DPB of the memory 360. The various filtering methods may include,for example, deblocking filtering, sample adaptive offset, an adaptiveloop filter, a bilateral filter or the like.

The (modified) reconstructed picture which has been stored in the DPB ofthe memory 360 may be used as a reference picture in the inter predictor332. The memory 360 may store motion information of a block in thecurrent picture, from which motion information has been derived (ordecoded) and/or motion information of blocks in an already reconstructedpicture. The stored motion information may be transmitted to the interpredictor 260 to be utilized as motion information of a neighboringblock or motion information of a temporal neighboring block. The memory360 may store reconstructed samples of reconstructed blocks in thecurrent picture, and transmit them to the intra predictor 331.

In this specification, the examples described in the predictor 330, thedequantizer 321, the inverse transformer 322, and the filter 350 of thedecoding apparatus 300 may be similarly or correspondingly applied tothe predictor 220, the dequantizer 234, the inverse transformer 235, andthe filter 260 of the encoding apparatus 200, respectively.

As described above, prediction is performed in order to increasecompression efficiency in performing video coding. Through this, apredicted block including prediction samples for a current block, whichis a coding target block, may be generated. Here, the predicted blockincludes prediction samples in a space domain (or pixel domain). Thepredicted block may be indentically derived in the encoding apparatusand the decoding apparatus, and the encoding apparatus may increaseimage coding efficiency by signaling to the decoding apparatus notoriginal sample value of an original block itself but information onresidual (residual information) between the original block and thepredicted block. The decoding apparatus may derive a residual blockincluding residual samples based on the residual information, generate areconstructed block including reconstructed samples by adding theresidual block to the predicted block, and generate a reconstructedpicture including reconstructed blocks.

The residual information may be generated through transform andquantization procedures. For example, the encoding apparatus may derivea residual block between the original block and the predicted block,derive transform coefficients by performing a transform procedure onresidual samples (residual sample array) included in the residual block,and derive quantized transform coefficients by performing a quantizationprocedure on the transform coefficients, so that it may signalassociated residual information to the decoding apparatus (through abitstream). Here, the residual information may include valueinformation, position information, a transform technique, transformkernel, a quantization parameter or the like of the quantized transformcoefficients. The decoding apparatus may perform aquantization/dequantization procedure and derive the residual samples(or residual sample block), based on residual information. The decodingapparatus may generate a reconstructed block based on a predicted blockand the residual block. The encoding apparatus may derive a residualblock by dequantizing/inverse transforming quantized transformcoefficients for reference for inter prediction of a next picture, andmay generate a reconstructed picture based on this.

FIG. 4 schematically illustrates a multiple transform techniqueaccording to an embodiment of the present disclosure.

Referring to FIG. 4, a transformer may correspond to the transformer inthe encoding apparatus of foregoing FIG. 2, and an inverse transformermay correspond to the inverse transformer in the encoding apparatus offoregoing FIG. 2, or to the inverse transformer in the decodingapparatus of FIG. 3.

The transformer may derive (primary) transform coefficients byperforming a primary transform based on residual samples (residualsample array) in a residual block (S410). This primary transform may bereferred to as a core transform. Herein, the primary transform may bebased on multiple transform selection (MTS), and when a multipletransform is applied as the primary transform, it may be referred to asa multiple core transform.

The multiple core transform may represent a method of transformingadditionally using discrete cosine transform (DCT) type 2 and discretesine transform (DST) type 7, DCT type 8, and/or DST type 1. That is, themultiple core transform may represent a transform method of transforminga residual signal (or residual block) of a space domain into transformcoefficients (or primary transform coefficients) of a frequency domainbased on a plurality of transform kernels selected from among the DCTtype 2, the DST type 7, the DCT type 8 and the DST type 1. Herein, theprimary transform coefficients may be called temporary transformcoefficients from the viewpoint of the transformer.

In other words, when the conventional transform method is applied,transform coefficients might be generated by applying transform from aspace domain to a frequency domain for a residual signal (or residualblock) based on the DCT type 2. Unlike to this, when the multiple coretransform is applied, transform coefficients (or primary transformcoefficients) may be generated by applying transform from a space domainto a frequency domain for a residual signal (or residual block) based onthe DCT type 2, the DST type 7, the DCT type 8, and/or DST type 1.Herein, the DCT type 2, the DST type 7, the DCT type 8, and the DST type1 may be called a transform type, transform kernel or transform core.

For reference, the DCT/DST transform types may be defined based on basisfunctions, and the basis functions may be represented as in thefollowing table.

TABLE 1 Transform Type Basis function T_(i)(j), i, j = 0, 1 . . . , N-1DCT-II${T_{i}(j)} = {\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot {\cos\left( \frac{\pi \cdot i \cdot \left( {{2j} + 1} \right)}{2N} \right)}}$${{where}\mspace{14mu}\omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.$ DCT-V${T_{i}(j)} = {\omega_{0} \cdot \omega_{1} \cdot \sqrt{\frac{2}{{2N} - 1}} \cdot {{\cos\left( \frac{2{\pi \cdot i \cdot j}}{{2N} - 1} \right)}.}}$${{where}\mspace{14mu}\omega_{0}} = \left\{ {{\begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix}.\mspace{14mu}\omega_{1}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {j = 0} \\1 & {j \neq 0}\end{matrix} \right.} \right.$ DCT-VIII${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\cos\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2j} + 1} \right)}{{4N} + 2} \right)}}$DST-I${T_{i}(j)} = {\sqrt{\frac{2}{N + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {i + 1} \right) \cdot \left( {j + 1} \right)}{N + 1} \right)}}$DST-VII${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}}$

If the multiple core transform is performed, then a vertical transformkernel and a horizontal transform kernel for a target block may beselected from among the transform kernels, a vertical transform for thetarget block may be performed based on the vertical transform kernel,and a horizontal transform for the target block may be performed basedon the horizontal transform kernel. Here, the horizontal transform mayrepresent a transform for horizontal components of the target block, andthe vertical transform may represent a transform for vertical componentsof the target block. The vertical transform kernel/horizontal transformkernel may be adaptively determined based on a prediction mode and/or atransform index of a target block (CU or sub-block) including a residualblock.

Further, according to an example, if the primary transform is performedby applying the MTS, a mapping relationship for transform kernels may beset by setting specific basis functions to predetermined values andcombining basis functions to be applied in the vertical transform or thehorizontal transform. For example, when the horizontal transform kernelis expressed as trTypeHor and the vertical direction transform kernel isexpressed as trTypeVer, a trTypeHor or trTypeVer value of 0 may be setto DCT2, a trTypeHor or trTypeVer value of 1 may be set to DST7, and atrTypeHor or trTypeVer value of 2 may be set to DCT8.

In this case, MTS index information may be encoded and signaled to thedecoding apparatus to indicate any one of a plurality of transformkernel sets. For example, an MTS index of 0 may indicate that bothtrTypeHor and trTypeVer values are 0, an MTS index of 1 may indicatethat both trTypeHor and trTypeVer values are 1, an MTS index of 2 mayindicate that the trTypeHor value is 2 and the trTypeVer value. Is 1, anMTS index of 3 may indicate that the trTypeHor value is 1 and thetrTypeVer value is 2, and an MTS index of 4 may indicate that both bothtrTypeHor and trTypeVer values are 2.

The transformer may derive modified (secondary) transform coefficientsby performing the secondary transform based on the (primary) transformcoefficients (S420). The primary transform is a transform from a spatialdomain to a frequency domain, and the secondary transform refers totransforming into a more compressive expression by using a correlationexisting between (primary) transform coefficients. The secondarytransform may include a non-separable transform. In this case, thesecondary transform may be called a non-separable secondary transform(NSST), or a mode-dependent non-separable secondary transform (MDNSST).The non-separable secondary transform may represent a transform whichgenerates modified transform coefficients (or secondary transformcoefficients) for a residual signal by secondary-transforming, based ona non-separable transform matrix, (primary) transform coefficientsderived through the primary transform. At this time, the verticaltransform and the horizontal transform may not be applied separately (orhorizontal and vertical transforms may not be applied independently) tothe (primary) transform coefficients, but the transforms may be appliedat once based on the non-separable transform matrix. In other words, thenon-separable secondary transform may represent a transform method inwhich the vertical and horizontal components of the (primary) transformcoefficients are not separated, and for example, two-dimensional signals(transform coefficients) are re-arranged to a one-dimensional signalthrough a certain determined direction (e.g., row-first direction orcolumn-first direction), and then modified transform coefficients (orsecondary transform coefficients) are generated based on thenon-separable transform matrix. For example, according to a row-firstorder, M×N blocks are disposed in a line in an order of a first row, asecond row, . . . , and an Nth row. According to a column-first order,M×N blocks are disposed in a line in an order of a first column, asecond column, . . . , and an Nth column. The non-separable secondarytransform may be applied to a top-left region of a block configured with(primary) transform coefficients (hereinafter, may be referred to as atransform coefficient block). For example, if the width (W) and theheight (H) of the transform coefficient block are all equal to orgreater than 8, an 8×8 non-separable secondary transform may be appliedto a top-left 8×8 region of the transform coefficient block. Further, ifthe width (W) and the height (H) of the transform coefficient block areall equal to or greater than 4, and the width (W) or the height (H) ofthe transform coefficient block is less than 8, then a 4×4 non-separablesecondary transform may be applied to a top-left min(8,W)×min(8,H)region of the transform coefficient block. However, the embodiment isnot limited to this, and for example, even if only the condition thatthe width (W) or height (H) of the transform coefficient block is equalto or greater than 4 is satisfied, the 4×4 non-separable secondarytransform may be applied to the top-left min(8,W)×min(8,H) region of thetransform coefficient block.

Specifically, for example, if a 4×4 input block is used, thenon-separable secondary transform may be performed as follows.

The 4×4 input block X may be represented as follows.

$\begin{matrix}{X = \begin{bmatrix}X_{00} & X_{01} & X_{02} & X_{03} \\X_{10} & X_{11} & X_{12} & X_{13} \\X_{20} & X_{21} & X_{22} & X_{23} \\X_{30} & X_{31} & X_{32} & X_{33}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

If the X is represented in the form of a vector, the vector

may be represented as below.

=[X ⁰⁰ X ⁰¹ X ⁰² X ⁰³ X ¹⁰ X ¹¹ X ¹² X ¹³ X ²⁰ X ²¹ X ²² X ²³ X ³⁰ X ³¹X ³² X ³³]^(T)  [Equation 2]

In Equation 2, the vector

is a one-dimensional vector obtained by rearranging the two-dimensionalblock X of Equation 1 according to the row-first order.

In this case, the secondary non-separable transform may be calculated asbelow.

=T·

  [Equation 3]

In this equation,

represents a transform coefficient vector, and T represents a 16×16(non-separable) transform matrix.

Through foregoing Equation 3, a 16×1 transform coefficient vector

may be derived, and the

may be re-organized into a 4×4 block through a scan order (horizontal,vertical, diagonal and the like). However, the above-describedcalculation is an example, and hypercube-Givens transform (HyGT) or thelike may be used for the calculation of the non-separable secondarytransform in order to reduce the computational complexity of thenon-separable secondary transform.

Meanwhile, in the non-separable secondary transform, a transform kernel(or transform core, transform type) may be selected to be modedependent. In this case, the mode may include the intra prediction modeand/or the inter prediction mode.

As described above, the non-separable secondary transform may beperformed based on an 8×8 transform or a 4×4 transform determined basedon the width (W) and the height (H) of the transform coefficient block.The 8×8 transform refers to a transform that is applicable to an 8×8region included in the transform coefficient block when both W and H areequal to or greater than 8, and the 8×8 region may be a top-left 8×8region in the transform coefficient block. Similarly, the 4×4 transformrefers to a transform that is applicable to a 4×4 region included in thetransform coefficient block when both W and H are equal to or greaterthan 4, and the 4×4 region may be a top-left 4×4 region in the transformcoefficient block. For example, an 8×8 transform kernel matrix may be a64×64/16×64 matrix, and a 4×4 transform kernel matrix may be a16×16/8×16 matrix.

Here, to select a mode-based transform kernel, three non-separablesecondary transform kernels may be configured per transform set for thenon-separable secondary transform for both the 8×8 transform and the 4×4transform, and there may be 35 transform sets. That is, 35 transformsets may be configured for the 8×8 transform, and 35 transform sets maybe configured for the 4×4 transform. In this case, three 8×8 transformkernels may be included in each of the 35 transform sets for the 8×8transform, and three 4×4 transform kernels may be included in each ofthe 35 transform sets for the 4×4 transform. The sizes of thetransforms, the numbers of sets, and the numbers of transform kernels ineach set mentioned above are merely for illustration. Instead, a sizeother than 8×8 or 4×4 may be used, n sets may be configured, and ktransform kernels may be included in each set.

The transform set may be called an NSST set, and the transform kernel inthe NSST set may be called an NSST kernel. The selection of a specificset from among the transform sets may be performed, for example, basedon the intra prediction mode of the target block (CU or sub-block).

For reference, as an example, the intra prediction mode may include twonon-directional (or non-angular) intra prediction modes and 65directional (or angular) intra prediction modes. The non-directionalintra prediction modes may include a No. 0 planar intra prediction mode,and a No. 1 DC intra prediction mode, and the directional intraprediction modes may include 65 intra prediction modes between a No. 2intra prediction mode and a No. 66 intra prediction mode. However, thisis an example, and the present disclosure may be applied to a case wherethere are different number of intra prediction modes. Meanwhile,according to circumstances, a No. 67 intra prediction mode may befurther used, and the No. 67 intra prediction mode may represent alinear model (LM) mode.

FIG. 5 illustrates directional intra modes of 65 prediction directions.

Referring to FIG. 5, on the basis of the No. 34 intra prediction modehaving a left upward diagonal prediction direction, the intra predictionmode having a horizontal directionality and the intra prediction modehaving vertical directionality may be classified. H and V of FIG. 5 meanhorizontal directionality and vertical directionality, respectively, andnumerals −32 to 32 indicate displacements in 1/32 units on the samplegrid position. This may represent an offset for the mode index value.The Nos. 2 to 33 intra prediction modes have the horizontaldirectionality, and the Nos. 34 to 66 intra prediction modes have thevertical directionality. Meanwhile, strictly speaking, the No. 34 intraprediction mode may be considered as being neither horizontal norvertical, but it may be classified as belonging to the horizontaldirectionality in terms of determining the transform set of thesecondary transform. This is because the input data is transposed to beused for the vertical direction mode symmetrical on the basis of the No.34 intra prediction mode, and the input data alignment method for thehorizontal mode is used for the No. 34 intra prediction mode.Transposing input data means that rows and columns of two-dimensionalblock data M×N are switched into N×M data. The No. 18 intra predictionmode and the No. 50 intra prediction mode may represent a horizontalintra prediction mode and a vertical intra prediction mode,respectively, and the No. 2 intra prediction mode may be called a rightupward diagonal intra prediction mode because it has a left referencepixel and predicts in a right upward direction. In the same manner, theNo. 34 intra prediction mode may be called a right downward diagonalintra prediction mode, and the No. 66 intra prediction mode may becalled a left downward diagonal intra prediction mode.

In this case, mapping between the 35 transform sets and the intraprediction modes may be, for example, represented as in the followingtable. For reference, if an LM mode is applied to a target block, thesecondary transform may not be applied to the target block.

TABLE 2 intra mode  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17set  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 intra mode 3435 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 set 34 33 32 31 30 2928 27 26 25 24 23 22 21 20 19 18 17 intra mode 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 set 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3233 intra mode 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 (LM) set16 15 14 13 12 11 10  9  8  7  6  5  4  3  2 NULL

Meanwhile, if a specific set is determined to be used, one of ktransform kernels in the specific set may be selected through thenon-separable secondary transform index. The encoding apparatus mayderive a non-separable secondary transform index indicating a specifictransform kernel based on the rate-distortion (RD) check, and may signalthe non-separable secondary transform index to the decoding apparatus.The decoding apparatus may select one from among k transform kernels inthe specific set based on the non-separable secondary transform index.For example, the NSST index value 0 may indicate a first non-separablesecondary transform kernel, the NSST index value 1 may indicate a secondnon-separable secondary transform kernel, and the NSST index value 2 mayindicate a third non-separable secondary transform kernel.Alternatively, the NSST index value 0 may indicate that the firstnon-separable secondary transform is not applied to a target block, andthe NSST index values 1 to 3 may indicate the three transform kernels.

Referring back to FIG. 4, the transformer may perform the non-separablesecondary transform based on the selected transform kernels, and mayobtain modified (secondary) transform coefficients. As described above,the modified transform coefficients may be derived as transformcoefficients quantized through the quantizer, and may be encoded andsignaled to the decoding apparatus and transferred to thedequantizer/inverse transformer in the encoding apparatus.

Meanwhile, as described above, if the secondary transform is omitted,(primary) transform coefficients, which are an output of the primary(separable) transform, may be derived as transform coefficientsquantized through the quantizer as described above, and may be encodedand signaled to the decoding apparatus and transferred to thedequantizer/inverse transformer in the encoding apparatus.

The inverse transformer may perform a series of procedures in theinverse order to that in which they have been performed in theabove-described transformer. The inverse transformer may receive(dequantized) transformer coefficients, and derive (primary) transformcoefficients by performing a secondary (inverse) transform (S450), andmay obtain a residual block (residual samples) by performing a primary(inverse) transform on the (primary) transform coefficients (S460). Inthis connection, the primary transform coefficients may be calledmodified transform coefficients from the viewpoint of the inversetransformer. As described above, the encoding apparatus and the decodingapparatus may generate the reconstructed block based on the residualblock and the predicted block, and may generate the reconstructedpicture based on the reconstructed block.

The decoding apparatus may further include a secondary inverse transformapplication determinator (or an element to determine whether to apply asecondary inverse transform) and a secondary inverse transformdeterminator (or an element to determine a secondary inverse transform).The secondary inverse transform application determinator may determinewhether to apply a secondary inverse transform. For example, thesecondary inverse transform may be an NSST or an RST, and the secondaryinverse transform application determinator may determine whether toapply the secondary inverse transform based on a secondary transformflag obtained by parsing the bitstream. In another example, thesecondary inverse transform application determinator may determinewhether to apply the secondary inverse transform based on a transformcoefficient of a residual block.

The secondary inverse transform determinator may determine a secondaryinverse transform. In this case, the secondary inverse transformdeterminator may determine the secondary inverse transform applied tothe current block based on an NSST (or RST) transform set specifiedaccording to an intra prediction mode. In an embodiment, a secondarytransform determination method may be determined depending on a primarytransform determination method. Various combinations of primarytransforms and secondary transforms may be determined according to theintra prediction mode. Further, in an example, the secondary inversetransform determinator may determine a region to which a secondaryinverse transform is applied based on the size of the current block.

Meanwhile, as described above, if the secondary (inverse) transform isomitted, (dequantized) transform coefficients may be received, theprimary (separable) inverse transform may be performed, and the residualblock (residual samples) may be obtained. As described above, theencoding apparatus and the decoding apparatus may generate thereconstructed block based on the residual block and the predicted block,and may generate the reconstructed picture based on the reconstructedblock.

Meanwhile, in the present disclosure, a reduced secondary transform(RST) in which the size of a transform matrix (kernel) is reduced may beapplied in the concept of NSST in order to reduce the amount ofcomputation and memory required for the non-separable secondarytransform.

Meanwhile, the transform kernel, the transform matrix, and thecoefficient constituting the transform kernel matrix, that is, thekernel coefficient or the matrix coefficient, described in the presentdisclosure may be expressed in 8 bits. This may be a condition forimplementation in the decoding apparatus and the encoding apparatus, andmay reduce the amount of memory required to store the transform kernelwith a performance degradation that can be reasonably accommodatedcompared to the existing 9 bits or 10 bits. In addition, the expressingof the kernel matrix in 8 bits may allow a small multiplier to be used,and may be more suitable for single instruction multiple data (SIMD)instructions used for optimal software implementation.

In the present specification, the term “RST” may mean a transform whichis performed on residual samples for a target block based on a transformmatrix whose size is reduced according to a reduced factor. In the caseof performing the reduced transform, the amount of computation requiredfor transform may be reduced due to a reduction in the size of thetransform matrix. That is, the RST may be used to address thecomputational complexity issue occurring at the non-separable transformor the transform of a block of a great size.

RST may be referred to as various terms, such as reduced transform,reduced secondary transform, reduction transform, simplified transform,simple transform, and the like, and the name which RST may be referredto as is not limited to the listed examples. Alternatively, since theRST is mainly performed in a low frequency region including a non-zerocoefficient in a transform block, it may be referred to as aLow-Frequency Non-Separable Transform (LFNST).

Meanwhile, when the secondary inverse transform is performed based onRST, the inverse transformer 235 of the encoding apparatus 200 and theinverse transformer 322 of the decoding apparatus 300 may include aninverse reduced secondary transformer which derives modified transformcoefficients based on the inverse RST of the transform coefficients, andan inverse primary transformer which derives residual samples for thetarget block based on the inverse primary transform of the modifiedtransform coefficients. The inverse primary transform refers to theinverse transform of the primary transform applied to the residual. Inthe present disclosure, deriving a transform coefficient based on atransform may refer to deriving a transform coefficient by applying thetransform.

FIG. 6 is a diagram illustrating an RST according to an embodiment ofthe present disclosure.

In the present specification, the term “target block” may mean a currentblock or a residual block on which coding is performed.

In the RST according to an example, an N-dimensional vector may bemapped to an R-dimensional vector located in another space, so that thereduced transform matrix may be determined, where R is less than N. Nmay mean the square of the length of a side of a block to which thetransform is applied, or the total number of transform coefficientscorresponding to a block to which the transform is applied, and thereduced factor may mean an R/N value. The reduced factor may be referredto as a reduced factor, reduction factor, simplified factor, simplefactor or other various terms. Meanwhile, R may be referred to as areduced coefficient, but according to circumstances, the reduced factormay mean R. Further, according to circumstances, the reduced factor maymean the N/R value.

In an example, the reduced factor or the reduced coefficient may besignaled through a bitstream, but the example is not limited to this.For example, a predefined value for the reduced factor or the reducedcoefficient may be stored in each of the encoding apparatus 200 and thedecoding apparatus 300, and in this case, the reduced factor or thereduced coefficient may not be signaled separately.

The size of the reduced transform matrix according to an example may beR×N less than N×N, the size of a conventional transform matrix, and maybe defined as in Equation 4 below.

$\begin{matrix}{T_{RxN} = \begin{bmatrix}t_{11} & t_{12} & t_{13} & \ldots & t_{1N} \\t_{21} & t_{22} & t_{23} & \; & t_{2N} \\\; & \vdots & \; & \ddots & \vdots \\t_{R1} & t_{R\; 2} & T_{R\; 3} & \ldots & t_{RN}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

The matrix Tin the Reduced Transform block shown in FIG. 6A may mean thematrix TR×N of Equation 4. As shown in FIG. 6A, when the reducedtransform matrix TR×N is multiplied to residual samples for the targetblock, transform coefficients for the target block may be derived.

In an example, if the size of the block to which the transform isapplied is 8×8 and R=16 (i.e., R/N=16/64=¼), then the RST according toFIG. 6A may be expressed as a matrix operation as shown in Equation 5below. In this case, memory and multiplication calculation can bereduced to approximately ¼ by the reduced factor.

$\begin{matrix}{\begin{bmatrix}t_{1,1} & t_{1,2} & t_{1,3} & \ldots & t_{1,64} \\t_{2,1} & t_{2,2} & t_{2,3} & \; & t_{2,64} \\\; & \vdots & \; & \ddots & {\;\vdots} \\t_{16,1} & t_{16,2} & t_{16,3} & \ldots & t_{16,64}\end{bmatrix} \times \begin{bmatrix}r_{1} \\r_{2} \\\vdots \\r_{64}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In Equation 5, r₁ to r₆₄ may represent residual samples for the targetblock and may be specifically transform coefficients generated byapplying a primary transform. As a result of the calculation of Equation5, transform coefficients c_(i) for the target block may be derived, anda process of deriving c_(i) may be as in Equation 6.

$\begin{matrix}{{{for}\mspace{14mu} i\mspace{14mu}{from}\mspace{14mu}{to}\mspace{14mu} R\text{:}}\mspace{20mu}{{ci} = 0}\mspace{20mu}{{for}\mspace{14mu} j\mspace{14mu}{from}\mspace{14mu} 1\mspace{14mu}{to}\mspace{14mu} N}\mspace{40mu}{{{ci}+={ti}},{j*{rj}}}} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

As a result of the calculation of Equation 6, transform coefficients c₁to c_(R) for the target block may be derived. That is, when R=16,transform coefficients c₁ to c₁₆ for the target block may be derived.If, instead of RST, a regular transform is applied and a transformmatrix of 64×64 (N×N) size is multiplied to residual samples of 64×1(N×1) size, then only 16 (R) transform coefficients are derived for thetarget block because RST was applied, although 64 (N) transformcoefficients are derived for the target block. Since the total number oftransform coefficients for the target block is reduced from N to R, theamount of data transmitted by the encoding apparatus 200 to the decodingapparatus 300 decreases, so efficiency of transmission between theencoding apparatus 200 and the decoding apparatus 300 can be improved.

When considered from the viewpoint of the size of the transform matrix,the size of the regular transform matrix is 64×64 (N×N), but the size ofthe reduced transform matrix is reduced to 16×64 (R×N), so memory usagein a case of performing the RST can be reduced by an R/N ratio whencompared with a case of performing the regular transform. In addition,when compared to the number of multiplication calculations N×N in a caseof using the regular transform matrix, the use of the reduced transformmatrix can reduce the number of multiplication calculations by the R/Nratio (R×N).

In an example, the transformer 232 of the encoding apparatus 200 mayderive transform coefficients for the target block by performing theprimary transform and the RST-based secondary transform on residualsamples for the target block. These transform coefficients may betransferred to the inverse transformer of the decoding apparatus 300,and the inverse transformer 322 of the decoding apparatus 300 may derivethe modified transform coefficients based on the inverse reducedsecondary transform (RST) for the transform coefficients, and may deriveresidual samples for the target block based on the inverse primarytransform for the modified transform coefficients.

The size of the inverse RST matrix T_(N×R) according to an example isN×R less than the size N×N of the regular inverse transform matrix, andis in a transpose relationship with the reduced transform matrix T_(R×N)shown in Equation 4.

The matrix Tt in the Reduced Inv. Transform block shown in FIG. 6B maymean the inverse RST matrix T_(R×N) ^(T) (the superscript T meanstranspose). When the inverse RST matrix T_(R×N) ^(T) is multiplied tothe transform coefficients for the target block as shown in FIG. 6B, themodified transform coefficients for the target block or the residualsamples for the target block may be derived. The inverse RST matrixT_(R×N) ^(T) may be expressed as (T_(R×N) ^(T))N×R.

More specifically, when the inverse RST is applied as the secondaryinverse transform, the modified transform coefficients for the targetblock may be derived when the inverse RST matrix T_(R×N) ^(T) ismultiplied to the transform coefficients for the target block.Meanwhile, the inverse RST may be applied as the inverse primarytransform, and in this case, the residual samples for the target blockmay be derived when the inverse RST matrix T_(R×N) ^(T) is multiplied tothe transform coefficients for the target block.

In an example, if the size of the block to which the inverse transformis applied is 8×8 and R=16 (i.e., R/N=16/64=¼), then the RST accordingto FIG. 6B may be expressed as a matrix operation as shown in Equation 7below.

$\begin{matrix}{\begin{bmatrix}t_{1,1} & \; & t_{2,1} & \; & t_{16,1} \\t_{1,2} & \; & t_{2,2} & {\ldots\;} & t_{16,2} \\t_{1,3} & \; & t_{2,3} & \; & t_{16,3} \\{\;\vdots} & \; & \vdots & \; & {\;\vdots} \\\; & \vdots & \; & \ddots & \vdots \\t_{1,64} & \; & t_{2,64} & \ldots & t_{16,64}\end{bmatrix} \times \begin{bmatrix}c_{1} \\c_{2} \\\vdots \\c_{64}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

In Equation 7, c₁ to c₁₆ may represent the transform coefficients forthe target block. As a result of the calculation of Equation 7, r_(j)representing the modified transform coefficients for the target block orthe residual samples for the target block may be derived, and theprocess of deriving r_(j) may be as in Equation 8.

$\begin{matrix}{{{for}\mspace{14mu} i\mspace{14mu}{from}\mspace{14mu}{to}\mspace{14mu} N\text{:}}\mspace{20mu}{{ri} = 0}\mspace{20mu}{{for}\mspace{14mu} j\mspace{14mu}{from}\mspace{14mu} 1\mspace{14mu}{to}\mspace{14mu} R}\mspace{40mu}{{{ri}+={tj}},{i*{cj}}}} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

As a result of the calculation of Equation 8, r₁ to r_(N) representingthe modified transform coefficients for the target block or the residualsamples for the target block may be derived. When considered from theviewpoint of the size of the inverse transform matrix, the size of theregular inverse transform matrix is 64×64 (N×N), but the size of thereduced inverse transform matrix is reduced to 64×16 (R×N), so memoryusage in a case of performing the inverse RST can be reduced by an R/Nratio when compared with a case of performing the regular inversetransform. In addition, when compared to the number of multiplicationcalculations N×N in a case of using the regular inverse transformmatrix, the use of the reduced inverse transform matrix can reduce thenumber of multiplication calculations by the R/N ratio (N×R).

A transform set configuration shown in Table 2 may also be applied to an8×8 RST. That is, the 8×8 RST may be applied according to a transformset in Table 2. Since one transform set includes two or three transforms(kernels) according to an intra prediction mode, it may be configured toselect one of up to four transforms including that in a case where nosecondary transform is applied. In a transform where no secondarytransform is applied, it may be considered to apply an identity matrix.Assuming that indexes 0, 1, 2, and 3 are respectively assigned to thefour transforms (e.g., index 0 may be allocated to a case where anidentity matrix is applied, that is, a case where no secondary transformis applied), an NSST index as a syntax element may be signaled for eachtransform coefficient block, thereby designating a transform to beapplied. That is, through the NSST index, it is possible to designate an8×8 NSST for a top-left 8×8 block and to designate an 8×8 RST in an RSTconfiguration. The 8×8 NSST and the 8×8 RST refer to transformsapplicable to an 8×8 region included in the transform coefficient blockwhen both W and H of the target block to be transformed are equal to orgreater than 8, and the 8×8 region may be a top-left 8×8 region in thetransform coefficient block. Similarly, a 4×4 NSST and a 4×4 RST referto transforms applicable to a 4×4 region included in the transformcoefficient block when both W and H of the target block to are equal toor greater than 4, and the 4×4 region may be a top-left 4×4 region inthe transform coefficient block.

If the (forward) 8×8 RST illustrated in Equation 4 is applied, 16significant transform coefficients are generated. Thus, it is consideredthat 64 pieces of input data forming the 8×8 region is reduced to 16pieces of output data, and only ¼ of the region is filled withsignificant transform coefficients from the perspective of atwo-dimensional region. Accordingly, the 16 pieces of output dataobtained by applying the forward 8×8 RST may fill the top-left region(transform coefficients 1 to 16) of the block as shown in FIG. 7.

FIG. 7 is a diagram illustrating a transform coefficient scanning orderaccording to an embodiment of the present disclosure. As describedabove, when the forward scanning order starts from a first transformcoefficient, reverse scanning may be performed in directions and ordersindicated by arrows shown in FIG. 7 from 64th to 17th transformcoefficients in the forward scanning order.

In FIG. 7, a top-left 4×4 region is a region of interest (ROI) filledwith significant transform coefficients, and the remaining region isempty. The empty region may be filled with 0s by default.

That is, when an 8×8 RST with a 16×64 forward transform matrix isapplied to the 8×8 region, output transform coefficients may be arrangedin the top-left 4×4 region, and the region where no output transformcoefficient exists may be filled with 0s (from the 64th to 17thtransform coefficients) according to the scanning order of FIG. 7.

If a non-zero significant transform coefficient is found outside the ROIof FIG. 7, it is certain that the 8×8 RST has not been applied, and thusNSST index coding may be omitted. On the contrary, if a non-zerotransform coefficient is not found outside the ROI of FIG. 7 (e.g., if atransform coefficient is set to 0 in a region other than the ROI in acase where the 8×8 RST is applied), the 8×8 RST is likely to have beenapplied, and thus NSST index coding may be performed. This conditionalNSST index coding may be performed after a residual coding processbecause it is necessary to check the presence or absence of a non-zerotransform coefficient.

The present disclosure discloses methods for optimizing a design and anassociation of an RST that can be applied to a 4×4 block from an RSTstructure described in this embodiment. Some concepts can be applied notonly to a 4×4 RST but also to an 8×8 RST or other types of transforms.

FIG. 8 is a flowchart illustrating an inverse RST process according toan embodiment of the present disclosure.

Each operation disclosed in FIG. 8 may be performed by the decodingapparatus 300 illustrated in FIG. 3. Specifically, S800 may be performedby the dequantizer 321 illustrated in FIG. 3, and S810 and S820 may beperformed by the inverse transformer 322 illustrated in FIG. 3.Therefore, a description of specific details overlapping with thoseexplained above with reference to FIG. 3 will be omitted or will be madebriefly. In the present disclosure, an RST may be applied to a transformin a forward direction, and an inverse RST may mean a transform appliedto an inverse direction.

In an embodiment, the specific operations according to the inverse RSTmay be different from the specific operations according to the RST onlyin that their operation orders are opposite to each other, and thespecific operations according to the inverse RST may be substantiallysimilar to the specific operations according to the RST. Accordingly, aperson skilled in the art will readily understand that the descriptionsof S800 to S820 for the inverse RST described below may be applied tothe RST in the same or similar manner.

The decoding apparatus 300 according to an embodiment may derive thetransform coefficients by performing dequantization on the quantizedtransform coefficients for the target block (S800).

The decoding apparatus 300 may determine whether to apply an inversesecondary transform after an inverse primary transform and before theinverse secondary transform. For example, the inverse secondarytransform may be an NSST or an RST. For example, the decoding apparatusmay determine whether to apply the inverse secondary transform based ona secondary transform flag parsed from a bitstream. In another example,the decoding apparatus may determine whether to apply the inversesecondary transform based on a transform coefficient of a residualblock.

The decoding apparatus 300 may determine an inverse secondary transform.In this case, the decoding apparatus 300 may determine the secondaryinverse transform applied to the current block based on an NSST (or RST)transform set specified according to an intra prediction mode. In anembodiment, a secondary transform determination method may be determineddepending on a primary transform determination method. For example, itmay be determined to apply an RST or LFNST only when DCT-2 is applied asa transform kernel in the primary transform. Alternatively, variouscombinations of primary transforms and secondary transforms may bedetermined according to the intra prediction mode.

Further, in an example, the decoding apparatus 300 may determine aregion to which the inverse secondary transform is applied based on thesize of the current block before determining the inverse secondarytransform.

The decoding apparatus 300 according to an embodiment may select atransform kernel (S810). More specifically, the decoding apparatus 300may select the transform kernel based on at least one of informations ona transform index, a width and height of a region to which the transformis applied, an intra prediction mode used in image decoding, and a colorcomponent of the target block. However, the example is not limited tothis, and for example, the transform kernel may be predefined, andseparate information for selecting the transform kernel may not besignaled.

In one example, information on the color component of the target blockmay be indicated through Cldx. If the target block is a luma block, Cldxmay indicate 0, and if the target block is a chroma block, for example,a Cb block or a Cr block, then Cldx may indicate a non-zero value (forexample, 1).

The decoding apparatus 300 according to an embodiment may apply theinverse RST to transform coefficients based on the selected transformkernel and the reduced factor (S820).

Hereinafter, a method for determining a secondary NSST set, that is, asecondary transform set or a transform set, in view of an intraprediction mode and the size of a block according to an embodiment ofthe present disclosure is proposed.

In an embodiment, a set for a current transform block may be configuredbased on the intra prediction mode described above, thereby applying atransform set including transform kernels having various sizes to thetransform block. Transform sets in Table 3 are expressed using 0 to 3 asin Table 4.

TABLE 3 Intra mode  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17NSST Set  0  0  2  2  2  2  2  2  2  2  2  2  2 18 18 18 18 18 Intramode 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 NSST Set 3434 34 34 34 34 34 34 34 34 34 18 18 18 18 18 18 18 Intra mode 18 19 2021 22 23 24 25 26 27 28 29 30 31 32 33 NSST Set 18 18 18 18 18 18 34 3434 34 34 34 34 34 34 34 Intra mode 52 53 54 55 56 57 58 59 60 61 62 6364 65 66 NSST Set 18 18 18 18  2  2  2  2  2  2  2  2  2  2  2

TABLE 4 Intra mode  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17NSST Set  0  0  1  1  1  1  1  1  1  1  1  1  1  2  2  2  2  2 Intramode 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 NSST Set  3 3  3  3  3  3  3  3  3  3  3  2  2  2  2  2  2  2 Intra mode 18 19 2021 22 23 24 25 26 27 28 29 30 31 32 33 NSST Set  2  2  2  2  2  2  3  3 3  3  3  3  3  3  3  3 Intra mode 52 53 54 55 56 57 58 59 60 61 62 6364 65 66 NSST Set  2  2  2  2  1  1  1  1  1  1  1  1  1  1  1

Indexes 0, 2, 18, and 34 illustrated in Table 3 correspond to 0, 1, 2,and 3 in Table 4, respectively. In Table 3 and Table 4, only fourtransform sets are used instead of 35 transform sets, therebysignificantly reducing memory space.

Various numbers of transform kernel matrices that may be included ineach transform set may be set as shown in the following tables.

TABLE 5 0 NSST Set (DC, Planar) 1 2 3 # of transform 2 2 2 2 kernels

TABLE 6 0 NSST Set (DC, Planar) 1 2 3 # of transform 2 1 1 1 kernels

TABLE 7 0 NSST Set (DC, Planar) 1 2 3 # of transform 1 1 1 1 kernels

According to Table 5, two available transform kernels are used for eachtransform set, and accordingly a transform index ranges from 0 to 2.

According to Table 6, two available transform kernels are used fortransform set 0, that is, a transform set according to a DC mode and aplanar mode among intra prediction modes, and one transform kernel isused for each of the remaining transform sets. Here, an availabletransform index for transform set 1 ranges from 0 to 2, and a transformindex for the remaining transform sets 1 to 3 ranges from 0 to 1.

According to Table 7, one available transform kernel is used for eachtransform set, and accordingly a transform index ranges from 0 to 1.

In transform set mapping of Table 3, a total of four transform sets maybe used, and the four transform sets may be rearranged to bedistinguished by indexes 0, 1, 2, and 3 as shown in Table 4. Table 8 andTable 9 illustrate four transform sets available for secondarytransform, wherein Table 8 presents transform kernel matrices applicableto an 8×8 block, and Table 9 presents transform kernel matricesapplicable to a 4×4 block. Table 8 and Table 9 include two transformkernel matrices per transform set, and two transform kernel matrices maybe applied to all intra prediction modes as shown in Table 5.

TABLE 8 const int g_alNSST8x8[4][2][16][64] = {  { //0   {    {−118, 22,21, −3, 4, 1, 2, 1, 31, −17, −3, 0, −1, 0, 0, 0, 16, 0, −4, 0, −1, 0, 0,0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, −1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},    {−22, −88, −46,0, 9, 0, 3, 0, −60, 29, −7, 3, 0, 1, 0, 0, 35, 2, −8, −1, −2, 0, −1, 0,−3, 3, 0, 0, 0, 0, 0, 0, 0, 7, 2, −2, −1, 0, 0, 0, 0, −1, 1, 0, 0, 0, 0,0, 0, 3, 1, −1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},    {−10, 68,−13, −7, −3, −2, −1, −1, −90, 1, 32, −1, 4, 0, 2, 0, 32, −32, −3, 4, −1,1, 0, 0, 7, 2, −5, 0, 0, 0, 0, 0, 6, −3, 0, 0, 0, 0, 0, 0, 1, 0, −1, 0,0, 0, 0, 0, 2, −1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0},    {16,−16, 12, 1, −1, 0, 0, 0, −10, −111, 20, 18, 4, 4, 2, 1, 20, 24, −33, −1,0, 0, 0, 0, −4, 17, 1, −7, 0, −1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, −1,3, 0, −1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, −1, 1,0, 0, 0, 0, 0, 0},    {−31, −40, −94, 48, −5, 10, −2, 4, −25, −13, 15,−14, 6, −2, 2, −1, −18, 17, 19, −9, −1, −1, 0, −1, 15, −5, −3, 2, −1, 0,0, 0, 0, 4, 2, −2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, −1,0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0},    {13, 3, −42, 11, −2, 3, −1, 1,31, 16, −15, −5, −2, −1, −1, 0, 99, 4, −10, −5, −4, −1, −2, −1, −49, 10,13, 0, 2, 0, 1, 0, 6, −5, 3, 0, 0, 0, 0, 0, −9, 1, 2, 0, 0, 0, 0, 0, 3,−1, 1, 0, 0, 0, 0, 0, 0,−4, 0, 1, 0, 0, 0, 0, 0},    {3, −32, −2, −2, 2,0, 1, 0, 37, −2, 70, −15, −3, −3, −1, −1, 9, −81, −2, 28, −2, 4, 0, 2,8, 20, −36, −1, 3, −1, 0, 0, −5, 4, 3, −4, 1, 0, 0, 0, 1, 3, −4, 0, 0,0, 0, 0, −1, 0, 1, 0, 0, 0, 0, 0, 1, −2, 0, 0, 0, 0, 0},    {0, 6, 13,−7, −1, 0, 0, 10, 30, 89, −20, −3, −5, 8, −2, 6, 75, −13, 4, −3, −2, −1,−1, −5, −27, 0, 5, 3, 1, 1, 1, −1, −3, −7, 2, 1, 1, 0, 0, 0, −4, −1, 1,0, 0, 0, 0, 0, 1, −1, 1, 0, 0, 0, 0, 0, −2, −1, 0, 0, 0, 0, 0},    {−13,−18, −41, −108, 26, −10, 7, −3, −8, −12, −4, −7, −11, 3, −1, 1, −5, 8,21, 20, −9, 0, −1, 0, −15, −3, 5, 4, 2, −1, 0, 0, 5, −1, 0, 3, −1, 0, 0,0, −2, 0, 1, 0, 0, 0, 0, 0, 1, −1, 0, 0, 0, 0, 0, −2, −1, 0, 0, 0, 0,0},    {9, 2, −10, −18, 4, −1, 1, 0, 13, 8, −16, 0, −2, 1, 0, 0, 44, 19,−11, −3, −4, −1, −1, 0, 107, 9, −17, −2, −5, −1, −2, 0, −27, 10, 9, 0,1, 0, 0, 0, 10, −3, 0, 1, 0, 0, 0, 0, −7, 1, 2, 1, 0, 0, 0, 0, 3, −1, 0,0, 0, 0, 0},    {12, 1, 27, 11, 0, 0, 0, 0, 2, −33, −6, −43, 8, −2, 2,0, 28, −4, 83, −5, −16, 0, −2, 0, 5, −48, −15, 36, 1, 2, 0, 1, 2, 6,−18, −6, 4, 0, 1, 0, −1, −2, 2, 2, −1, 0, 0, 0, 1, 2, −2, 1, 0, 0, 0, 0,0, 0, 0, 1, 0, 0, 0, 0},    {0, 3, 4, 0, 9, −4, 2, −1, 4, 7, 21, 68,−14, 5, −3, 2, 7, 18, 66, −37, 7, −4, 1, −1, −5, 47, −42, −14, 4, 0, 0,0, −7, −21, −1, 9, −1, 1, 0, 0, −3, 6, −3, −2, 1, 0, 0, 0, 2, −3, 1, 1,0, 0, 0, 0, −1, 2, −1, −1, 0, 0, 0, 0},    {0, −8, −14, −2, −2, −1, −1,0, 8, 0, 6, 78, −8, 3, −3, 1, 11, −20, −13, 5, 18, −1, 2, 0, 6, −88, −3,0, 2, 6, 0, 2, 5, 13, −23, −4, −1, 0, 0, 0, −1, −3, 5, −6, −1, 0, 0, 0,1, 3, −3, 0, 0, 0, 0, 0, 0, −1, 1, −2, 0, 0, 0, 0},    {5, 3, −3, −1, 2,0, 0, 0, 8, 3, −1, −9, −1, 1, 0, 0, −2, 2, −9, −3, 2, −1, 1, 0, 28, 17,−1, 2, −1, −1, 0, 0, 108, 25, −36, −6, −4, −1, −2, 0, −32, 2, 13, −2, 0,0, 0, 0, 11, −1, −3, 1, 0, 0, 0, 0, −4, 1, 1, 0, 0, 0, 0, 0},    {6, 9,−1, 31, 108, −27, 11, −4, −1, −1, 1, −2, −31, 15, −4, 2, −4, −4, −11,−6, −33, 3, 0, 0, 1, −8, 6, −1, 10, −4, 0, 0, −2, 2, 1, 0, −11, 0, 0, 1,0, 0, 0, 1, 0, 0, 0, 0, 0, −1, 0, −1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0},   {1, −8, 3, −12, 1, −1, 0, 7, −4, 25, −6, 0, 0, 1, −1, 3, −27, 10,−72, 87, −1, 1, 0, 16, −7, 77, −8, −32, −3, −5, 3, 1, 1, −7, 3, 42, −7,−3, 0, −1, 0, −2, −3, 2, 5, −1, 1, 0, 0, −2, 0, 4, −1, 0, 0, 0, 1, 0,−1, 0, 1, 0, 0, 0},   },   {    {108, −48, −10, −1, −2, −1, −1, 0, −45,6, 9, 1, 1 ,0, 1, −9, 10, 1, −1, 0, 0, 0, 0, −3, 1, −1, 0, 0, 0, 0, 0,−2, 1, 0, 0, 0, 0, 0, 0, −1, 1, 0, 0, 0, 0, 0, 0, −1, 1, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0},    {54, 60, −38, −6, −, −1, −2, 0, 71, −32,−20, 5, −1, 1, −1, 0, −34, −19, 15, 5, 1, 1, 1, 0, −8, 4, 5, −2, 0, 0,0, 0, −7, 0, 2, 0, 0, 0, 0, 0, −1, 1, 1, 0, 0, 0, 0, 0, −2, −1, 1, 0, 0,0, 0, 0, −1, 0, 0, 0, 0, 0, 0, 0},    {−3, −88, 29, 12, 5, 1, 2, 1, 83,−1, −7, −5, −5, −1, −1, 0, −16, 6, −1, −2, 1, 0, 0, 0, −12, 6, 2, 0, 1,0, 0, 0, −5, 5, −1, −1, 0, 0, 0, 0, −2, 1, 0, 0, 0, 0, 0, 0, −1, 1, 0,0, 0, 0, 0, 0, −1, 0, 0, 0, 0, 0, 0, 0},    {25, 24, 34, −2, 0, −4, 0,−2, 20, 87, −33, −20, −2, −3, −1, −1, 29, −29, −38, 10, 3, 2, 0, 1, −15,−20, 8, 11, 0, 1, 0, 0, −1, −4, 3, 2, −1, 0, 0, 0, −5, −2, 2, 1, 0, 0,0, 0, 0, −1, 0, 0, 0, 0, 0, 0, −1, −1, 1, 0, 0, 0, 0, 0},    {7, 23, 93,−32, −3, −6, 1, −3, −14, −19, 10, −14, 3, 0, 1, 0, −68, 0, 0, 5, 7, 1,1, 0, 16, 16, −7, 0, 0, −1 ,0 ,0 ,3 ,0 ,−8 ,1 ,0, 0, 0, 0 ,4 ,1 ,−2 ,1,0, 0, 0, 0, 0, −1, −2, 1, 0, 0, 0, 0, 2, 0, −1, 0, 0, 0, 0, 0},    {22,9, 46, −17, −2, −3, 0, −1, 10, −70, 5, 7, 4, 2, 1, 1, 86, −2, −19, 0,−3, 0, −1, 0, −21, 0, 1, 0, 1, 0, 0, 0, −5, 5, 0, 0, 1, 0, 0, 0, −4, 2,0, −1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, −2, 1, 0, 0, 0, 0, 0, 0},   {−12, −27, −1, −15, 8, −2, 2, −1, −30, −16, −70, 25, 7, 3, 1, 2, −4,−68, 27, 35, −4, 3, 0, 1, −8, 22, 32, −11, −8, −1, −1, −1, 5, 6, −3, −8,−1, 1, 0, 0, 0, 3, 2, −2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,−1, 0, 0, 0, 0},    {−2, −6, −12, −53, 10, −3, 3, −1, 4, −5, −68, 6, 15,0, 2, 1, 6, 77, −5, 13, −4, −3, −1, −1, 40, −6, −17, 3, −6, 1, −1, 0,−3, 12, 4, 7, −1, 1, 0, 0, 2, −1, 3, 0, −1, 0, 0, 0, −2, 0, 2, 1, 0, 0,0, 0, −1, 1, 0, 0, 0, 0, 0},    {6, 6, 13, 86, −12, 2, −5, 2, −14, −28,−50, −16, −8, 4, 1, 1, −12, 2, −52, −3, 14, 3, 1, 1, 29, −17, 4, 19, −2,−1, −1, 0, −1, −4, 11, −2, −2, 1, 0, 0, 1, 0, 3, 0, 0, −1, 0, 0, −2, 1,1, −1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0},    {16, 22, 37, 52, 0, −2,−2, 0, 10, 33, −14, 44, −11, 0, −2, 1, 25, 25, 58, −10, −20, −1, −2, −1,23, 30, −5, −36, −3 ,−1 ,0 ,−1, 2, −6, −15, −6, 4, 1, 1, 0, −1, 0, 0,−2, 1, 1, 0, 0, 0, −2, −1, −1, 1, 0, 0, 0, 0, 0, 0, −1, 0, 0, 0, 0},   {8, −13, −8, −13, 0, 0, 0, 0, −18, −1, 28, −5, 0, −1, 0, 0, 32, −51,−5, 10, 2, 2, 0, 1, 104, 4, −24, 0, −3, −1, −2, 0, 8, −10, −5, 2, 0, 0,0, 0, −2, 0, 4, 0, 0, 0, 0, 0, −4, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,0, 0, 0},    {0, 0, −8, 0, −27, 5, −3, 1, −4, −1, −18, −96, −10, 11, 2,0, 22, 6, 35, −8, 28, −1, −1, −1, −1, 50, 1, −6, −4, −10, 0, −1, 2, 1,−20, 3, 4, −1, 1, 0, −1, 0, 2, 6, 0, −1, 0, 0, 1, −2, −2, 1, 1, 0, 0, 0,−1, 0, 1, 2, 0, 0, 0, 0},    {−3, −7, 0, −10, −2, 1, 1, 0, −9, −5, −30,11, −14, 4, −3, 1, −5, −29, 6, −66, −9, 5, 0, 1, −13, −7, −62, −3, 51,2, 3, 1, −1, −9, 2, 49, 8, −9, 2, −1, 0, 0, 5, 2, −8, −1, −1, 0, 0, 0−3,1, 4, 0, −1, 0, 0, 0, 1, 1, 0, −2, 0, 1, 0},    {1, −5, 16, −1, −18, 3,−1, 0, −2, −8, −3, −24, −11, 7, 0, 0, 4, −3, 66, −1, 1, 1, −2, 0, 10,−95, 21, 2, 0, 0, 0, 2, −12, −17, 23, −3, 3, 0, 1, 0, 3, 4, −2, −2, 2,−1, 0, 0, −3, 3, 2, 0, 0, 0, 0, 0, 1, 0, −1, −1, 1, 0, 0, 0},    {−6,−7, −2, −100, 25, −3, 3, 1, 6, 1, 41, −34, 20, −5, 1, −2, 6, −16, 14,34, −1, −3, 0, 0, 3, 1, −11, 7, −6, 0, 1, −1, 2, −1, −6, 5, −2, 0, 0, 0,0, 1, −2, 2, −1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0},   {−1, −1, 1, 22, 3, −6, −2, 0, −1, 0, 5, 0, 16, −2, 3, 0, 1, −4, 13,75, 29, −1, 1, −1, −28, −16, −75, −2, −14, −4, 1, 0, −14, −23, −25, 20,1, 4, 1, 1, 1, 3, 13, 3, −6, 0, 0, 0, 0, −2, −1, 0, 0, 0, 0, 0, −1, 0,1, 0, −2, 0, 0,0},   }  },  { //1   {    {−98, 54, −1, 4, 0, 1, 0, 1,56, −9, −13, 0, −2, 0, −1, 0, 3, −18, 3, 3, 0, 0, 0, 0, 2, 0, 5, −1, 0,0, 0, 0, 0, −2, 0, −1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, −1, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},    {64, 9, −18, 0, −4, 0, −1, 0, 86,−48, −13, 2, −2, 0, −1, 0, −28, −22, 17, 3, 2, 1, 1, 0, −11, 8, 8, −3,0, −1, 0, 0, −10, 3, 0, −1, 1, 0, 0, 0, −3, 3, 1, 0, 0, 0, 0, −4, 1, 0,0, 0, 0, 0, 0, −2, 1, 0, 0, 0, 0, 0, 0},    {−17, 18, −11, 4, −3, 2, −1,1, −29, −65, 40, 2, 5, 1, 2, 0, −54, 53, 32, −15, 2, −3, 1, −1, 16, 27,−24, −13, 1, −1, 0, 0, 0, 1, −12, 4, 4, 0, 1, 0, 6, 1, −3,−3, 0, 0, 0,0, 0, 0, −1, 1, 0, 0, 0, 0, 3, 0, −1, 1, −1, 0, 0, 0},    {−35, −98, 50,−1, 8, 0, 3, 0, 38, −9, 21, −9, 0, −2, 0, −1, −8, 3, 7, −8, 1, −1, 0, 0,−11, 19, 1, −6, 2, −1, 0, 0, −2, 0, −9, −2, 2, 0, 0, 0, −2, 4, −3, 2, 1,0, 0, 0, 0, 2, −2, 1, −1, −1, 0, 0, −1, 1, −1, 0, 0, 0, 0, 0},    {7, 0,−10, 1, −1, 0, 0, 0, 45, 8, 32, −15, 2, −4, 1, −2, 14, 61, −55, −19, 2,−3, 0, −1, 15, −40, −41, 30, 5, 3, 1, 1, −4, −13, 20, 19, −8, −1, −1, 0,2, −6, 6, −4, −8, 2, 0, 0, −3, −2, 4, −3, 1, 3, 0, 0, 0, −1, 1, 0, 1, 0,−1, 0},    {−3, 9, −20, 8, −4, 3, −1, 1, −7, −59, 30, 12, 0, 3, 0, 1,96, −9, 24, −15, −6, −2, −1, −1, −10, −22, 2, −16, 4, 0, 1, 0, −6, 6, 6,−2, 8, 0, 1, 0, −5, 4, 0, −1, 2, −2, 0, −1,−1, 3, 0, 0, 1, −1, 0, 0, −2,1, −1, −1, 0, 0, 0, 0},    {−19, −25, −25, 13, 0, 3, 0, 1, −25, −43, 4,6, 2, 0, 1, 0, −26, −23, −29, 22, 4, 3, 1, 1, −7, −36, 39, 44, −5, 3,−1, 1, 2, 18, 49, −8, −20, −3, −3, −1, 4, 7, 4, −22,−8, 2, 0, 0, 0, 4,1, −10, 2, 3, 1, 0, 1, 2, 0, −4, 3, −5, 0, 2, 0},    {4, 7, −11, 9, −2,4, −1, 1, 10, 10, 37, −19, −2, −4, 0, −1, −1, 15, −25, 8, 1, 3, −1, 1,27, −10, 49, −11, −19, 0, −4, 0, −6, 15, 21, −66, 16, 9, 2, 1, −7, −4,−31, 21, 43, −5, 2, 0, −2, 7, 0, 21, −6, −13, 0, −2, 0, −2, −4, 3, −5,0, 2, 0},    {6, 34, 16, −1, −1, 1, 0, 0, −5, 17, 58, −21, −1, −31, 1,−1, −11, −25, −12, −35, 7, −3, 2, −1, −80, 7, 1, 8, 19, 0, 4, 0 −5, 39,−2, 8, −2, −5, 0, −1, 1, 6, −13, 0, −5, 1, 0, 0, 4, 1, −5, 1, 0, 2, −1,0, 0, −2, −3, 1, 0, 0, −1, 0},    {−16, −34, −96, 25, −6, 6, −3, 2, 12,46, 16, 28, −8, 3, −21, 1, −13, −2, 19, −16, −8, 0, −1, 0, −15, 12, −4,−9, 7, 1, 1, 0, 4, 0, −9, 5, 2, −1, 0, 0, 0, −4, 4, 4, −2, 0, 0, 0, 1,1, 2, −2, −1, 0, 0, 0, −1, −1, 0, 1, 0, 0, 0, 0},    {−1, −5, −23, 1, 5,−1, 0, 0, −5, −25, 5, −17, 9, −4, 1, −1, 26, −17, −44, 8, 24, −2, 4, 0,10, 69, −6, 4, 1, 1, −10, 0, −2, 22, 7, −39, 7, −27, −6, −3, −1, −10,−30, −6, 13, −9, 8, 2, 2, 2, −2, 10, 6, −2, 4, 0, 0, −1, −1, 4, 4, −2,1, −1, 0},    {10, 14, 16, 7, −5, −1, 0, −1, 17, 25, 27, 17, −7, 0, 0,−1, 19, 29, 34, 11, −11, −2, −1, −1, 17, 34, 29, 11, −18, −6, −3, −2,16, 17, 22, −20, −43, −10, −6, −2−2, −8, −11, −52, −28, 7, 1, 2, −2, −8,−11, −24, 19, 14, 4, 3, 0, 0, −3, −7, 11, 4, 0, 0},    {7, 3, 13, 7, −3,3, 0, 1, 0, 8, 54, −9, −8, 0, −1, 0, −14, −68, 6, −31, 1, 1, 1, 0, 65,−9, −2, −2, 14, 1, 2, 0, 26, −32, 4, 9, 1, −3, 0, −1, −2, −12, 21, −1,−4, 0, 0, 0, −2, 4, 3, −9, 1, 1, 0, −1, −1, 2, −2, 0. 2, 0, 0, 0},   {1, 1, −4, 21, −9, 0, −2, 1, 1, −1, 15, −32, 14, −2, 4, −2, −1, 6,−20, 48, −17, 0, 3, 1, 7, −11, 32, −47, 5, 6, 0, 1, −12, 20, −32, 36,20, −19, 4, −4, 8,−8, 19, 8, −51, 8, −3, 0, −1, −3, −2, −26, 13, 13, −2,3, 2, −2, 3, 5, 3, −4, −1, 0},    {1, 4, 0, 14, 0, 1, 0, 0, −1, −1, 23,−15, −8, −1, 1, −1, 6, −2, −5, 28, 6, 3, 1, −1, −37, 21, −9, 3, −54, 4,−8, 1, 0, −45, 1, −13, 6, 33, −2, 8, 22, 1, 61, −5, 24, −7, −5,−1, 4, 1,−6, −32, 0, −14, 1, −2, −4, −2, −3, 4, 6, −1, 4, 0},    {−3, 2, −6, −85,12, −7, 5, −4, 4, −3, 29, 59, 20, −7, 3, 0, 3, −9, 31, 17, −43, 4, −3,0, −1, 13, 2, −11, −4, 16, −2, 2, 2, 1, 0, 0, 9, 1, −1, 0, 1, −6, 7, 3,0, −1, −2, 0, 1, 0, 1, −4, −2, −1, 0, 1, 0, 0, 0, 3, 0, −1, 1, 0},   },  {    {70, −48, 6, −4, 1, −1, 1, 0, −75, 43, 2, 1, 0, 0, 0, 0, 33, −5,−12, 3, −2, 1, −1, 0, −9, −9, 9, 0, 0, 0, 0, 0, 3, 6, −2, −2, 0, 0, 0,0, −2, −2, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, −1, −1, 0, 0, 0, 0,0, 0},    {−81, 3, 14, 2, 3, 0, 1, 0, −13, 55, −19, 1, −3, 1, −1, 0, 53,−46, −4, 6, −2, 1, −1, 0, −18, 3, 20, −7, 2, −1, 1, 0, 7, 6, −12, −1, 1,−1, 0, 0, −2, −4, 3, 4, −1, 0, 0, 0, 3, 1, −1, −2, 0, 0, 0, 0, −1, −1,1, 0, 0, 0, 0, 0},    {48, −19, 12, −3, 1, −1, 0, 0, 54, 3, −29, 2, −4,1, −2, 0, −6, −62, 34, 2, 3, 0, 1, 0, −34, 49, 6, −15, 4, −3, 1, −1, 7,−4, −23, 9, −1, 1, 0, 0, −5, −1, 11, 2, −3, 1, 0, 0, 1, 0, −3, −3, 1, 0,0, 0, −1, −1, 1, 0, 0, 0, 0},    {31, 18, −23, 7, −3, 2, −1, 1, 50, −5,10, −7, −1, −2, 0, −1, 35, −20, −36, 10, −2, 2, −1, 1, −2, −52, 50, 3,−2, 1, 0, 0, −22, 40, 0, −27, 7, −4, 2, −1, −1, −6, −20, 15, 3, 0, 1, 0,−3, 3, 9, 2, −6, 0, 0, 0, 0, −3, −2, −2, 0, 1, 0, 0},    {−19, −75, 28,1, 5, 9, 2, 9, 37, 42, 01, −14, 0, −3, 0, −1, −54, 12, −29, 18, −1, 3,−1, 1, 35, −9, 18, 1, −7, 2, −2, 1, −10, −5, −2, −11, 6, −1, 1, 0, 2, 5,0, 6, 0, −1, 0, 0, −2, −1, −2, −1, −2, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0},   {17, 4, 16, −1, 1, −2, 1, −1, 43, 33, −28, 12, −4, 2, −2, 1, 42, 7,−2, −15, −3, −2, −1, −1, 11, −30, −45, 20, 1, 3, 0, 1, 4, −38, 54, 7,−7, 2, −1, 1, −11, 23, −6, −25 ,7, −3, 1, −1, 0, −10, −6, 14, 2, 0, 1,0, −2, 3, 6, −4, −3, 0, 0, 0},    {−17, −29, 9, −13, 4, −2, 2, −1, −18,−30, −3, 20, −3, 3, −1, 1, −14, −20, 25, −17, 5, −1, 2, 09, −21, 10, 12,23, 2, 1, −1, 0, 4, 17, 34, −47, −4, −2, 0, −1, −19, 29, −57, 7, 26, −5,5, −1, 10, −8, 9, 31, −18, −1, −1, 0, −1, 2, 0, −14, −5, 5, −1, 1},   {−19, −64, −9, 1, 0, 1, 0, 0, 30, −30, 47, −4, 4, −1, 2, 0, 42, 27,−13, −20, 1, −4, 0, −1, −52, 4, −20, 12, 4, 1, 1, 0, 16, 10, −3, 14, −9,2, −2, 1, 2, −17, 12, −6, −3, 2, −1, 1, −1, 0, 2, −8, 5, −1, 0, 0, −3,0, −5, 3, 2, −1, 0, 0},    {−10, −33, 22, 2, −2, −1, −1, 0, −6, −41,−21, −1, 4, 2, 1, 1, 29, 2, 44, 13, −7, 1, −2, 0, 25, −28, −1, −44, 9,−4, 3, −1, −48, 26, 0, 30, 12, −3, 3, −1, 10, 20, −8,−10, −24, 2, −4, 1,−4, −13, −8, 5, 12, 6, 0, 2, 7, 2, 15, −3, −2, −4, −1, −1},    {5, 5, 6,−8, 2, −1, 1, 0, 15, 20, 0, 0, 2, −1, 0, 0, 28, 41, 6, −13, 3, −2, 1,−1, 15, 25, −32, −15, 1, −5, 0−2, −21, −7, −55, −7, 11, 0, 3, 0, −9, −5,−29, 57, 14, 1, 3, 0, 13, −8, 25, 28, −26, −2, −3, 0, 3, −10, 7, 13,−10, 6, −1, 1},    {−3, −24, −13, 3, 1, 1, 0, 0, −4, −45, 2, 9, −4, 0,−1, 0, 38, −24, 7, 10, 2, 2, 1, 1, 63, 4, 18, 6, −13, −1, −4, 0, 14,−57, −7, −27, −1, 2, 0, 1, −15, −10, 28, 13, 16, −2, 4, −1, −1, 17, 6,−11, 10, −5, 0, −2, −6, 1, −11, 5, 3, 4, 1, 0},    {5, 15, 74, −17, 7,−6, 3, −2, −2, −34, −44, −13, 6, −2, 1, −1, 11, 37, −21, 39, −7, 3, −1,1, −33, −7, 16, −4, −16, 5, −4, 1, 30, −4, −2, −14, 9, 1, 0, 1, −4, −15,6, 5, 6, −5, 2, −1, −1, 9, 3, −3, −4, 0, 1, −1, −3, 1, −7, 2, 1, 1, 0,0},    {−9, −13, −18, −4, 0, −1, 0, 0, −18, −25, −33, −6, 1, −1, 0, 0,−28, −38, −36, −2, 4, 0, 0, 0, −35, −44, −30, 13, 9, 3, 2, 1, −28, −31,−6, 30, 10, 4, 3, 1, −8, −7, 19, 35, 4, 1, 1, 0, 20, 0, 26, 16, −11, −3,−2, −1, 0, −2, 13, −5, −9, 1, −1, 0},    {0, 0, −3, 6, −10, 3, −2, 1,−1, −2, 1, −11, 14, 4, −2, −2, −1, −1, −6, 11, −14, 4, −2, 2, 4, 2, 8,11, 17, −3, 2, −1, −1, 3, −14, 8, −27, 1, −4, 1, 1, −18, 8, −40, 46, 2,7, 0, −15, 30, −31, 65, −2, 22, 4, −5, 10, −16, 18, −9, −44, 17, −3, 2},   {0, 4, −18, 25, 2, 1, −1, 0, −2, −7, 25, −35, −4, 0, 1, 0, 9, 1, −22,27, 11, −3, 1, −1, −18, 17, 10, −21, −23, 5, −3, 1, 16, −33, 3, 1, 34,−1, 3, 0, −30, 51, −7, 0, −20, −9, 0, −2, 11, −19, −30, 18, 3, 12, −2,2, 6, −9, 40, −9, −7, −3, −3, 1},    {−7, −18, −74, 0, −9, −1, −3, 0, 5,18, −49, 28, −2, 6, −1, 2, −4, 46, 32, 32, −9, 3, −2, 1, −30, −9, 17,−17, −15, 2, −3, 0, 7, −11, 4, −16, 6, 2, 0, 1, 6, −7, 5, −2, 6, −4, 2,−1, −7, 0, 3, −1, −2, −1, 2, −1, −3, 4, −1, −1, 0, 1, 0, 0},   }  },  {//2   {    {117, −39, −4, −5, −1, −2, 0, −1, 23, −4, −2, 0, −1, 0, 0, 0,−21, 6, 1, 1, 0, 0, 0, 0, −11, 3, 1, 0, 0, 0, 0, 0, −5, 2, 0, 0, 0, 0,0, 0, −3, 1, 0, 0, 0, 0, 0, 0, −2, 1, 0, 0, 0, 0, 0, 0, −1, 0, 0, 0, 0,0, 0, 0},    {22, −3, −2, −1, −1, 0, 0, 0, −116, 29, −5, 3, 1, 1, 1, 0,−27, −1, 3, 0, 1, 0, 0, 0, 24, −6, −1, 0, 0, 0, 0, 0, 12, −1, −1, 0, 0,0, 0, 0, 5, −1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,0, 0, 0},    {−21, 17, −4, 2, −1, 1, 0, 0, 23, 4, −3, 0, −1, 0, 0, 0,−118, 19, 7, 2, 2, 1, 1, 0, −18, −4, 3, 0, 1, 0, 0, 0, 19, −5, −1, 0, 0,0, 0, 0, 7, 01, −1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,0, 0, 0, 0},    {−34, −101, 54, −11, 9, −3, 4, −1, −7, −23, 10, −1, 1,0, 0, 0, −9, 28, −11, 3, −2, 1, −1, 0, 6, 8, −2, 0, −1, 0, 0, 0, 8, 1,−2, 0, 0, 0, 0, 2, 1, −1, 0, 0, 0, 0, 0, 1, 2, −1, 0, 0, 0, 0, 0, 1, 1,0, 0, 0, 0, 0, 0},    {6, 2, −3, 0, 0, 0, 0, 0, 28, −13, 1, −1, 0, −1,0, 0, −12, −14, 3, −1, 1, 0, 0, 0, 117, −16, −8, −1, −2, 0, −1, 0, 28,6, −4, 0, −1, 0, 0, 0, −12, 3, 1, 0, 1, 0, 0, 0, −5, 0, 1, 0, 0, 0, 0,0, 0, 1, 0, 0, 0, 0, 0, 0},    {6, 16, 1, −1, 0, 0, 0, 0, −23, −102, 36,−8, 6, −2, 2, −1, −14, −49, 10, 0, 1, 0, 1, 0, −12, 24, −9, 3, −1, 1, 0,0, 3, 17, −4, 1, −1, 0, 0, 0, 4, 5, −2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,0, 0, 0, 2, 0, 0, 0, 0, 0, 0},    {8, 0, 0, -1, 0, 0, 0, 0, 6, 2, -1, 0,0, 0, 0, 24, -7, 1, -1, 0, 0, 0, 0, -25, -11, 3, -1, 1, 0, 0, 0, 116,-14, -8, -1, -2, 0, -1, 0, 33, 2, -4, 0, -1, 0, 0, 0, 7, 2, 0, 0, 0, 0,0, 0, -1, 0, 0, 0, 0, 0, 0, 0},    {−9, −30, 18, −5, 2, −1, 1, −1, 12,44, 0, 0, −1, 0, 0, 0, −14, −99, 24, −4, 3, −1, 1, 0, −14, −31, −3, 2,0, 0, 0, 0, −8, 31, −7, 2, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0, 1, −2, 0,−1, 0, 0, 0, 0, 0, 1, −1, 0, 0, 0, 0, 0},    {−19, −51, −93, 27, −6, 5,−2, −2, −8, −21, −50, 15, −1, 2, −2, 1, −2, −8, 12, −4, 4, −1, 1, −1,−1, 3, 16, −4, 1, −1, 1, 0, 3, 8, 9, −2, 0, 0, 0, 0, 0, 3, 4, −1, −1, 0,0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0},    {2, −1, −1, 0,0, 0, 0, 0, 6, −3, −1, 0, 0, 0, 0, 0, 0, −2, 0, 0, 0, 0, 0, 0, 20, −31,1, 0, 0, 0, 0, 0, −33, −13, 3, 0, 1, 0, 0, 0, 117, −10, −9, 0, −2, 0,−1, 0, 24, 6, −3, 0, −1, 0, 0, 0, −5, 0, 1, 0, 0, 0, 0, 0},    {0, 5, 1,0, 0, 0, 0, 0, 5, 29, −10, 2, −1, 0, −1, 0, −2, 2, −10, 1, −1, 0, −1, 0,8, 94, −19, 1, −2, 0, −1, 0, 16, 72, 1, 0, 0, 0, 0, 0, 14, −7, 7, 1, 1,1, 0, 0, 2, −16, 2, −1, 0, 0, 0, 0, −1, −2, 0, 0, 0, 0, 0, 0},    {7,19, 48, −1, 2, 0, −9, −25, −87, 24, −6, 4, −4, 2, −6, −22, −51, 7, 0, 0,−1, 0, 0, −5, 29, −15, 4, −2, 1, −1, 1, 2, 18, −6, 1, −1, 0, 0, 0, 2, 7,2, −1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},    {−2,0, 1, 0, 0, 0, 0, 0, −2, 1, 1, 0, 0, 0, 0, 0, −2, −1, 0, 0, 0, 0, 0, 0,−2, 2, 1, 1, 0, 0, 0, 0, 0, −3, 1, 0, 0, 0, 0, 0, 16, 0, −1, 0, −1, 0,0, 0, −96, 6, 6, 0, 2, 0, 1, 0, −82, 0, 6, −1, 2, 0, 1, 0},    {2, 9,−1, 1, 1, 0, 0, 0, −4, −17, 1, 0, −1, 0, 0, 0, 7, 41, −1, 1, 0, 0, 0, 0,−9, −69, 3, −3, 1, −1, 0, 0, 0, 91, −13, 6, −1, 2, 0, 0, 13, 9, 7, 1, 0,1, 0, 0, −1, −25, 3, −4, 0, 0, 0, 0, −4, 3, −2, −1, 0, 0, 0, 0},    {−3,−7, −27, 5, 2, 0, −1, 0, 4, 9, 42, 7, 0, 1, 2, 0, −7, −14, −80, 17, −10,4, −3, 1, −5, −18, −39, −23, 1, −3, −1, −1, 4, 2, 59, −16, 3, −2, 1, 0,2, 4, 17, 5, 1, 0, 0, 0, 0, 2, −3, 3, 0, 0, 0, 0, 0, 0, −2, 1, 0, 0, 0,0},    {3, −2, 24, 69, −55, 22, −7, 4, 2, 1, 15, 26, −21, 7, −1, 1, 0,2, −3, −56, 39, −15, 5, −3, −1, 1, −13, −26, 15, −5, 0, −1, −1, −1, −6,1, −6, 1, −1, 0, 0, 0, −1, 1, −21, −1, 0, 0, −1, 0, 3, 0, 0, 0, 0, 0, 0,0, 0, 1, 1, 0, 0, 0},   },   {    {−72, 45, −8, 5, −1, 2, −1, 1, 77,−39, 2, −3, 0, −1, 0, 0, −37, 9, 5, −1, 1, 0, 0, 0, 11, 2, −4, 1, −1, 0,0, 0, −4, −2, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, −2, 0, 0, 0, 0,0, 0, 1, 0, 0, 0, 0, 0, 0, 0},    {−87, 11, 10, 0, 2, 0, 1, 0, −28, 33,−11, 3, −2, 1, −1, 0, 67, −31, 0, 0, −1, 0, 0, 0, −31, 5, 6, −1, 1, 0,1, 0, 9, 4, −3, 0, 0, 0, 0, 0, −2, −2, 0, 0, 0, 0, 0, 0, 3, −1, 0, 0, 0,0, 0, 0, −1, 0, 1, 0, 0, 0, 0, 0},    {19, −50, 33, −8, 5, −2, 2, −1,77, 29, −31, 2, −4, 0, −2, 0, −2, −40, 10, 2, 1, 0, 0, 0, −40, 26, 5,−2, 1, −1, 1, 0, 18, −3, −6, 2, −1, 0, 0, 0, −9 ,0, 3, 0, 0, 0, 0, 0, 3,0, −1, 0, 0, 0, 0, 0, −3, 0, 1, 0, 0, 0, 0, 0},    {−45, −54, 39, −8, 6,−2, 2, −1, −17, 44, −21, 1, −1, 0, −1, 0, −5, 0, 19, −6, 8, −2, 2, −1,1, 57, −19, 2, −2, −1, 0, 0, 0, −20, 0, 3, 0, 1, 0, 0, 0, 7, 2, 0, 0, 0,0, 0, −3, 0, −1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0},    {−5, 30, −8,1, −1, 0, 0, 0, −45, 14, −13, 8, −3, 2, −1, 1, −65, −39, 33, −7, 5, −1,2, 0, −4, 52, −17, −2, 0, −1, 0, 0, 37, −30, −5, 5, −2, 1, −1, 0, −20,6, 8, −2, 1, 0, 0, 0, 8, −1, −2, 0, 0, 0, 0, 0, −3, 1, 0, 0, 0, 0, 0,0},    {−23, −40, 10, −1, 2, 0, 1, 0, −16, −29, 25, −8, 4, −2, 2, −1,−22, 57, −25, 5, −2, 0, −1, 0, −56, −7, 4, 4, −1, 2, 0, 1, 61, −9, 0,−3, 0, −1, 0, 0, −27, 0, 3, 0, 1, 0, 1, 0, 11, 4, −1, 0, −1, 0, 0, 0,−5, −1, −1, 0, 0, 0, 0, 0},    {−5, −8, −52, 29, −8, 6, −3, 2, 10, 46,52, −28, 5, −5, 2, −1, −28, −36, −48, 15, 1, 1, −1, 1, −1, 2, 39, 1, −3,1, 0, 0, 2, 17, −15, −9, 3, −2, 0, −1, 0, −8, 1, 7, −2, 1, 0, 0, 1, 2,0, −4, 1, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0},    {−2, 10, 13, −7, 1, −1,1, 0, −23, −9, −16, 7, −1, 1, −1, 0, −47, −8, 10, −8, 0, 0, 0, 0, −70,−17, 13, 0, 4, −1, 1, 0, −36, 52, −4, 1, −1, 0, 0, 0, 51, −15, −10, 1,−2, 1, −1, 0, −22, −4, 5, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0},   {−23, −69, −31, 9, −2, 2, −2, 0, −6, −35, 41, −4, 0, 0, 1, 0, 8, −13,30, −17, 7, −3, 2, −1, −2, 44, −35, 4, 0, −1, 0, 0, −34, −12, 2, 8, −2,2, 0, 1, 30, 6, 1, −2, 0, 0, 0, 0, −12, −4, 1, 0, 1, 0, 0, 0, 3, 3, −1,0, 0, 0, 0, 0},    {1, 5, −53, 8, −3, 2, −3, 1, 19, 69, −2, 17, −6, 4,−1, 1, −3, 46, 37, −20, 2, −2, 1, −1, −28, −35, −39, 1, 2, −1, 0, 0, 0,−15, 12, 8, −2, 2, 0, 0, 5, 4, 4, −4, 1, −1, 0, 0, −5, 0, 1, 1, 0, 0, 0,0, −1, 1, −2, 0, 0, 0, 0, 0},    {6, 23, 19, −1, 1, 0, 1, 0, 1, 19, −2,−12, 5, −2, 1, −1, 1, 25, −46, 24, −6, 2, −2, 1, −27, 38, 4, −3, −7, 3,−2, 1, −35, −75, 20, −3, 5, −1, 1, 0, 37, 19, 3, −2, 1, −1, 1, 0, −20,−1, −1, 0, −1, 0, 0, 0, 1, 4, −2, 0, 0, 0, 0, 0},    {−3, −2, 0, 0, −1,0, 0, 0, −6, −5, 0, −1, 0, 0, 0, 0, −10, −5, −1, −1, 0, 0, 0, 0, −28,−7, 1, 0, 0, 0, 0, 0, −80, −7, 2, −1, 1, 0, 0, 0, −90, 8, 8, 1, 1, 0, 1,0, 23, 11, −3,−1, −1, 0, 0, 0, 2, −3, −1, 1, 0, 0, 0, 0},    {0, −3, −8,9, −4, 1, −2, 0, −2, −5, −8, −8, 5, 0, 2, 0, 6, 19, 26, 33, −13, 5, −2,1, −1, 1, 20, −70, 12, −6, 3, −2, −5, −8, −60, 43, 13, −2, 2, 0, 5, 1,35, 5, −23, 5, −2, 1, 0, −1, −8, −15, 13, 0, 0, 0, −1, 1, 0, 7, −2, −2,1, 0},    {−3, 0, −19, −62, 50, −15, 4, −4, 3, 1, 25, 61, −45, 10, −1,2, −2, −1, −26, −20, 10, 5, −4, 1, 4, 7, 24, −12, 14, −12, 5, −2, −4,−7, −15, 10, −11, 5, −1, 0, 2, 2, 5, 0, 2, 1, −1, 0, 1, 0, −1, −1, 0,01, 0, 0, −1, 0, 0, 1, 1, 0, 0, 0},    {0, −2, −3, 0, 1, 0, 0, 0, −1,−2, 4, −2, 0, 1, 0, 0, −4, −8, 4, 8, −2, 1, −1, 0, −4, −14, 9, −14, 5,−1, 1, 0, 1, 8, 14, 6, 5, −1, 1, −1, 2, 83, −36, −1, −9, 0, −2, 0, 28,−74, 0, 5, 1, 2, −1, 1, −17, 16, 14, −2, 1, −1, 1, 0},    {7, 26, 72, 0,1, 2, 4, 0, 7, 25, 83, −5, 3, 1, 4, 0, −2, −4, 0, −2, −4, 10, −18, 6,−4, 0, −1, −2, −8, −37, −5, 3, −3, 0, −1, −2, 8, −21, 15, −1, 2, 0, 0,0, 2, 6, 2, −3, 1, −1, 0, 3, −5, −2, −3, 2, 0, 0, 0, 0, 0, −1, 1, 0, 0,0, 0},   }  },  { //3   {    {−112, 43, 9, 3, 2, 1, 1, 0, 12, −31, 10,0, 2, 0, 1, 0, 23, −12, −8, 3, −1, 1, 0, 0, 4, 3, −4, −1, 0, 0, 0, 0, 4,0, −1, −1, 0, 0, 0, 0, 1, 1, −1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0},    {−21, −43, 19, 1, 3, 0, 1, 0, 91, −3, −38,2, −5, 1, −2, 0, −28, 51, 2, −9, 0, −2, 0, −1, −7, −9, 15, 1, 0, 0, 0,0, −3, 2, −2, 2, 0, 0, 0, 0, −1, −2, 2, 0, 0, 0, 0, 0, −1, 1, 0, 0, 0,0, 0, 0, −1, −1, 1, 0, 0, 0, 0, 0},    {23, −31, 17, −6, 4, −2, 1, −1,−3, −78, 10, 16, 0, 4, 0, 2, 34, 13, −64, 7, −1, 1, −1, 0, −13, 42, 1,−23, 2, −3, 0, −1, 2, −1, 12, 0, −4, 0, −1, 0, −3, 6, 0, 1, 1, −1, 0, 0,0, 0, 1, 0, 0, 1, −9, 0, −1, 2, 0, 0, 0, 0, 0, 0},    {−40, −93, 33, 8,5, 2, 2, 1, −53, 18, −22, 5, 2, 1, 0, 0, 13, −17, 9, −4, 1, 0, 0, 0, 16,−5, −16, 6, −1, 1, 0, 0, 1, 10, −6, −6, 2, −1, 0, 0, 3, 0, 0, −2, −2, 1,0, 0, 1, 2, 0, −1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0},    {−10, 19, −3,−2, −1, 0, 0, 0, −52, 2, −38, 9, 1, 2, 0, 1, 13, 56, −7, −45, 4, −7, 1,−3, 1, −11, 68, −10, −12, 1, −3, 0, 9, −17, 1, 27, −8, 2, −1, 1, −3,1,−2, 5, 3, −2, 1, −1, 2, −3, −1, 2, 1, 0, 0, 0, −1, 0, 1, 0, 1, 0, 1,0},    {12, 20, 61, −17, 3, −4, 1, −2, −21, −52, 9, 26, −1, 4, 0, 2,−56, 1, 4, 0, 4, 1, 1, 0, 16, −47, −3, 20, −2, 2, 0, 1, −1, 9, −37, −3,8, 0, 1, 0, 3, 0, 4, −12, −3, 1, 0, 0, 1, 3, −5, 2, −1, −1, 0, 0, 1, 0,1, −2, 0, 0, 0, 0},    {−17, −3, −6, 0, 4, 0, 1, 0, −8, 23, 7, 13, −4,3, −2, 1, −75, −19, −44, 6, 20, −1, 5, 0, 27, 8, 3, −61, −2, 1, −1, 0,4, −2, 33, 12, −37, −1, −3, 0, 4, 1, −7, 13, 7, −5, 0, −1, −1, −1, 4,−2, 1, 2, 1, 0, 2, −1, −2, 1, 0, −1, 1, 0},    {6, −24, 18, −17, 2, −3,2, −1, 24, −11, 56, −21, −6, −2, 0, −1, 18, −32, 28, −6, −8, 09, −1, 1,26, −13, 44, 2, −25, 3, −5, 1, 4, −15, 4, 56, −20, −1, −2, 1, −6, 8,−25, 21, 13, −5, 1, −1, −2, −1, −3, 1, 4, 1, 1, 0, −1, 1, −5, 3, 2, −2,2, 0},    {10, 29, 86, −25, 1, −4, 2, −2, 13, 62, −2, 27, −9, 2, −2, 1,37, −4, −19, 9, 9, −1, 1, 0, −16, 12, 0, −10, 3, 1, 1, 0, −4, −5, 10,−1, −2, −1, 0, 0, −1, −4, 1, 4, 0, 0, 0, 0, −1, −1, 0, 0, 0, 0, 0, 0,−1, −1, 0, 1, 0, 0, 0, 0},    {20, 18, 1, 12, −5, 4, −2, 1, 25, −17,−63, 16, 14, −1, 2, 0, 20, −57, 0, −48, 10, 0, 1, −1, 49, 10, −7, −3,−23, 5, −5, 1, −12, 26, 2, 5, −3, −3, 1, −1, 4, −1, 4, 1, 1, 0, 0, 0,−3, 3, 0, −1, 0, 0, 1, 0, 1, 0, 1, 0, −1, 0, 0, 0},    {9, 12, 22, 13,−4, −2, 1, 0, −9, −25, −25, −27, 6, 2, 0, 0, 12, 15, 35, 14, −11, −4,−1, −1, −12, −32, −29, −26, 17, 10, 3, 3, 21, 15, 36, −13, −53, −6, −6,−1,−5, 0, −2, 50, 16, −11, −3, −3, 1, 1, −4, −13, 5, 9, 2, 0, −1, −2, 1,9, −1, −5, 1, 0},    {3, −11, −31, 5, 9, −1, 0, 0, 15, 9, 10, 25, −8, 5,−1, 0, 38, −10, −24, 7, 41, −3, 7, −1, −14, −71, 0, 34, 2, 20, 0, 6, 6,0, −45, −2, −26, −2, −1, 0, −3, 10, 9, −14,−1, −5, −1, −3, 6, 1, −2, 4,−2, 1, 1, −1, −2, 3, 2, −4, 1, 0, 1, 0},    {−1, 4, −3, 14, 1, −3, 0,−1, −4, 4, −11, 9, 3, 2, −3, 1, −1, 7, −7, 22, 3, −7, 0, −3, −5, 8, −11,12, 7, 1, −1, 0, −7, 20, −15, 43, 13, −23, 4, −6, −3, −6, −15, −19, 85,−5, −5, 0, 2, −12, 3, −41, 4, 39, −8, 5, −2, 0, −5, −4, −9, 11, 3, −2},   {−2, −4, 6, 19, −5, 5, −1, 2, 1, −12, −24, 11, 16, −3, 4, −1, −22,−51, 6, −16, 6, 7, −1, 2, −82, −2, 5, 1, −6, 3, 2, 0, 10, −56, −2, 5, 1,−1, 1, 1, −1, 5, −33, 0, −1, 0, −1, 0, 1, 0, 0, −12, 1, −2, 1, −1, 0, 1,−2, −1, −3, 0, 0, 0},    {5, 2, 1, 23, 1, 1, 0, −1, 4, −4, −11, 17, 16,3, −3, 0, 9, −4, 5, 29, 8, −8, 4, −7, 33, −1, 21, 2, 71, −1, −1, 0, 1,−14, −2, 3, 8, 63, −2, 10, −10, 2, −39, 0, −12, 10, 23, −1, −2, −6, −2,−21, 2, −7, 7, 2, 0, 01, −6, −5, −4, 2, 1, 2},    {4, 2, 18, 86, −4, 5,−4, 4, 4, 6, 47, 33, 38, −9, 6, −1, 1, 29, 16, −22, 24, 16, −1, 3, 11,14, −9, 1, −19, 14, 3, 2, 5, −2, 5, −5, −3, −9, 5, −1, −2, 1, 3, 2, −3,−1, −2, 1, −1, −1, 1, 3, −1, 0, 0, 0, 0, 0, 0, 2, 0, −1, 1, 0},   },   {   {102, −43, −2, −4, −1, −1, 0, −1, 35, 10, −12, 1, −3, 0, −1, 0, −33,32, −1, 0, 0, 0, 0, 0, −17, 2, 8, −1, 1, 0, 0, 0, −6, −1, 3, 1, 0, 0, 0,0, −3, −1, 1, 1, 0, 0, 0, 0, −2, 0, 0, 0, 0, 0, 0, 0, −1, 0, 1, 0, 0, 0,0, 0},    {−51, 4, 9, 0, 2, 0, 1, 0, 100, −31, −11, −1, −3, 0, −1, 0,−5, 40, −9, 0, −2, 0, −1, 0, −26, 5, 12, −1, 2, 0, 1, 0, 0, −6, 2, 2, 0,0, 0, 0, −3, 0, 0, 0, 0, 0, 0, 0, 0, −1, 0, 0, 0, 0, 0, 0, −2. 0, 1, 0,0, 0, 0, 0},    {−26, 5, 3, 2, 0, 1, 0, 0, 7, 41, −11, 0, −2, 0, −1, 0,−88, 2, 33, −2, 6, −1, 2, 0, 30, −58, 2, 6, 0, 2, 0, 1, 14, 5, −18, 2,−1, 0, −1, 0, 3, 2, −1, −3, 1, −1, 0, 0, 1, 1, −1,−1, 0, 0, 0, 0, 2, −1,0, 0, 0, 0, 0, 0},    {−13, −72, 21, −1, 4, 0, 2, 0, 27, −14, −7, 0, −1,0, 0, 0, 7, −40, 0, 6, 0, 2, 0, 1, 49, 9, −47, 3, −5, 1, −2, 0, −15, 50,−1, −14, 0, −3, 0, −1, −2, −1, 14, −1, −1, 0, 0, 0, −3, 5, 0, 0, 0, 0,0, 0, 0, 0, 3, 0, −1, 0, 0, 0},    {34, 81, −28, 2, −6, 1, −2, 0, 42,−11, 21, −7, 3, −2, 1, −1, −22, −47, 5, 9, 0, 2, 0, 1, 15, 18, −30, −1,−2, 0, −1, 0, −13, 21, 13, −10, 0, −2, 0, −1, −1, −7, 10, 5, −1, 0, 0,0, −3, −1, −1, 2, 1, 0, 0, 0, 0, −1, 0, 0, 0, 0, 0, 0},    {23, 9, 8,−5, 1, −1, 1, −1, −4, −74, 18, 4, 3, 1, 1, 1, 7, −6, −4, 2, −1, 1, 0, 0,−1, −50, −3, 14, −1, 4, 0, 1, 28, 8, −61, 3, −2, 1, −1, 0, −10, 39, −2,−24, 2, −4, 1, −1, 4, 1, 13, −4, −4, 0, −1, 0, −2, 4, 2, 1, −1, 0, 0,0},    {−9, −6, 13, −2, 3, −1, 1, 0, −35, −63, 13, 8, 1, 2, 0, 1, −72,13, −45, 6, −1, 2, −1, 1, 19, 34, 4, −23, 3, −5, 1, −2, 2, −11, 29, 1,−1, 1, 0, 0, 7, −9, −4, 8, −2, 1, 0, 0, 0, 1, −4, 1, 1, 0, 0, 0, 2, 1,−2, −1, 1, 0, 0, 0},    {4, 32, 6, −3, 0, −2, 0, −1, 0, −24, −11, −39,6, −2, 2, −1, 1, 11, 65, 2, −14, −2, −3, 0, −1, 12, −10, −9, 4, 1, 0, 0,0, −1, 60, 5, −27, −1, −5, 0, −2, −12, −17, 52, 7, −2, 0, 0, 0, 4, −11,−10, 11, 1, 1, 0, 1, −3, 0, 6, −1, 1, −1, 0, 0},    {11, 14, −6, −3, 0,−1, 0, 0, 14, 0, −5, 2, −3, 1, −1, 0, 22, 19, −9, −17, 1, −4, 0, −1, 92,7, 51, −1, −8, 0, −2, 0, −29, −3, −8, 41, 3, 5, 1, 2, −8, 1, −21, −12,9, −1, 2, 0, −3, 0, 5, −5, −3, 0, −1, 0, 1, 1, −1, 0, 0, 0, 0, 0},   {13, −1, 32, −4, 1, −2, 1, −1, 4, −35, 10, 1, 3, −1, 1, 0, 17, −14,38, −3, 0, −1, 0, 0, 5, −43, 13, 14, 1, 1, 0, 1, 3, −16, 30, 9, −4, 2,−1, 1, 1, −45, −11, 52, −2, 7, −1, 3, −3, 6, −56, 9, 12, 0, 3, 0, −3, 7,−8, −9, 5, −1, 1, 0},    {7, 1, −5, −1, −2, 0, −1, 0, 13, 15, −3, −6,−1, −1, 0, −1, 16, 5, −12, 3, −3, 0, −1, 0, 32, 2, −2, −22, 4, −6, 1,−2, 94, 1, 32, −7, −12, 0, −4, 0, −7, 44, −1, 35, −6, 3, −1, 1, −16, 9,4, 5, 12, −1, 3, 0, −4, −6, 0, 1, 3, 1, 0, 0},    {13, 31, 99, −18, 11,−5, 5, −2, 9, 33, −9, 27, −6, 6, −2, 2, 4, −21, −44, 5, 4, 1, 1, 0, −6,−8, 1, −16, 2, −1, 0, 0, −5, −2, −6, −1, −2, 1, 0, 0, −2, 3, 1, −7, −1,−1, 0, −1, 0, −1, 4, 1, −2, 0, −1, 0, 0, 0, 0, 0, 0, 0, 0, 0},    {−2,−10, −5, −1, 2, −2, 0, −1, 6, 4, 20, −1, −4, 0, −1, 0, −1, −12, −5, 0,2, −2, 1, −1, 4, 0, 25, −3, −6, −1, −2, 0, −1, −12, −11, 2, 2, −3, 1,−1, 24, 7, 51, −9, −17, 0, −5, 0, −1, −12, 0, 87, −3, 2, 0, 1, −13, 6,−45, 0, 32, −2, 5, 0},    {7, 14, 22, 0, −2, 1, 0, 0, −5, −21, −25, −2,1, −1, 0, −1, 4, 13, 41, −7, −2, 0, 0, 0, 18, 13, −29, 12, 2, 1, 1, 1,−24, −47, 2, −28, −2, −3, 0, −1, 50, 31, −9, 5, −12, −1, −3, 0, −22, 25,26, 9, 5, −1, 1, 0, 4, −21, −4, 7, 4, 1, 1, 1},    {−1, −2, −9, −3, 0,−1, 1, 0, −1, 4, 17, 63, −3, 2, 0, 1, 0, 17, 49, −22, 44, −1, 6, 0, 2,24, −40, −38, −18, 18, −1, 4, 0, −31, −22, 2, −28, −10, 2, −2, 1, 2, −3,−4, 5, −9, −4, −1, −1, −2, 7, 0, 0, 2, −1, −1, 0, 2, −2, 0, 1, −1, 1, 0,0},    {2, −3, −44, 22, −4, 5, −2, 2, 5, −9, −64, 26, −2, 5, −2, 2, 10,−31, −48, −5, 13, −2, 3, −1, 1, −48, −1, −36, 14, −3, 2, −1, −13, −23,6, −19, −4, 3, −2, 1, −1, −12, −3, −2, −9, 3, −2, 0, 2, 2, −9, 3, −6, 1,−1, 0, 1, 5, −2, −1, −1, −1, 0, −1},   }  } };

TABLE 9 const int g_alNsst4x4[4][2][16][16] = {  { //0   {    {108, −44,−16, 2, −43, 19, 6, −1, −11, 6, 2, −1, 0, −1, 0, 0},    {37, 100, −55,−13, 2, −26, 14, 2, −14, −22, 13, 4, 1, 2, −2, 0},    {28, −21, −8, 6,102, −17, −31, 1, −53, 19, 14, −8, −8, 1, 4, 0},    {−33, −38, −94, 53,−5, −15, 29, −8, 7, 19, 19, −14, 3, 1, −4, 1},    {8, −11, 27, −7, −15,−105, 35, 25, −5, 37, −26, −5, 4, 17, −6, −6},    {−25, 1, 14, −2, −36,12, 15, −2, −98, 3, 29, 1, 55, −9, −20, 2},    {7, 7, 14, 2, 37, 33, 98,−37, −6, 12, −42, 10, −9, −15, −14, 9},    {0, 33, −7, −2, −12, 21, −26,1, 3, 100, −23, −27, 8, −45, 22, 10},    {16, 28, 39, 108, −5, −8, −15,−22, −5, −17, −16, −27, 4, 5, 3, 4},    {−10, −10, −34, 0, −8, 1, −25,6, −33, −26, −98, 32, 14, 5, 37, −16},    {−16, 3, 5, 4, −26, 5, 13, 11,−47, 1, 13, −7, −104, 3, 42, −4},    {3, 5, 1, 17, 16, 33, 26, 109, 3,−9, −9, −34, 10, −2, −9, −26},    {−3, 14, −5, −3, −3, 30, −10, −11, −6,40, −11, −5, −7, 108, −32, −22},    {4, 9, 11, 33, 1, 5, 8, 15, 11, 31,31, 99, 7, −6, 20, −50},    {3, −2, 8, −11, 10, 4, 28, −15, −4, 23, −34,42, 33, 101, −16},    {0, −2, −1, −13, −1, −7, −4, 35, −1, −8, −3, −38,−8, −31, −21, −109},   },   {    {−118, 32, 21, 3, 27, 4, −5, −2, 16,−3, −6, 0, 4, −2, 0, 0},    {−30, −97, 33, 16, −51, 3, 25, 3, 21, 30,−6, −8, 6, 3, −5, −2},    {0, 65, 20, −16, −99, 3, 35, 1, 10, −15, −5,4, 14, −1, −8, 0},    {24, 4, 63, −10, 21, −0, −5, −27, 14, −8, −40, −1,−10, −18, 0, 8},    {18, 5, 91, −2, 25, −74, 10, 17, −21, −12, −20, 4,−1, 13, −3, −4},    {8, −3, −1, −7, −24, −27, −90, 8, 69, −9, −38, 10,−2, 1, 20, 3},    {19, 30, −9, 5, 29, −15, 43, 5, 69, 75, −19, −29, −1,−5, −22, −6},    {−3, 21, 22, −3, −20, 9, −57, −9, −55, 86, 15, −22, 15,10, 7, −4},    {−7, −17, −1, 112, 7, −16, 9, −41, 8, 15, 17, 26, −2, 3,−1, 9},    {10, 5, 37, 0, 7, 22, −11, 36, 44, −4, 101, 10, −8, 15, 10,−19},    {2, 6, 11, 34, −10, −29, 1, −85, 12, −7, 36, −30, −46, −27, 27,32},    {8, −2, 2, 17, 11, 0, −4, −56, 17, 21, 9, −15, 86, 63, −20, −5},   {−4, 4, −8, −8, −4, 12, 16, −2, −2, −1, −22, −22, −54, 90, 53, −30},   {−4, −7, −1, −24, −7, −2, −24, 1, −4, −29, 5, −75, −34, −2, −78,−37},    {1, −6, 0, −30, 6, −1, 11, 25, 0, −18, 1, −79, 46, −36, 70, 0},   {1, 0, −1, 9, 0, −7, 3, −34, −1, 3, −5, 26, 8, −38, 20, −112},   } },  { //1   {    {−110, 40, 5, 3, 44, 13, −12 ,−1, 8, −15, −6, 2, 3,−2, 4, 2},    {−47, −29, 16, −1, −91, 42, 22, −2, 20, 40, −15, −5, 10,−5, −13, 1},    {36, 21, −3, −4, 17, 74, −32, −6, 58, −17, −49, 5, 1,−36, −2, 11},    {−13, −93, 27, 2, 49, −14, 33, −5, 51, −12, −1, −10,−1, −17, 2, −2},    {11, 28, −3, −2, 47, 3, 38, −10, 1, 85, −12, −27,17, 1, −58, −5},    {0, −35, 34, −4, 25, 58, −4, −16, −83, 2, −30, 1,−36, 23, 2, 12},    {−17, −47, −93, 16, 9, −5, −42, 30, −13, 28, −25,18, 0, −6, −12, 14},    {−5, 4, 20, −6, −15, −58, 3, −24, −31, −16, −68,3, 25, −59, −17, 38},    {6, −16, 34, −4, 9, −1, −49, −4, 3, 12, −7, 24,95, 51, 4, 2},    {−3, 2, 38, −15, −7, −41, −58, −5, 37, 32, −20, 17,−71, 31, 21, 4},    {−3, −5, −26, −9, −13, −1, 10, −18, 9, −64, −30,−38, 2, 58, −68, −19},    {5, 15, −19, 29, 2, −10, 49, −9, 22, 10, −53,8, −7, 59, 62, 40},    {−6, −8, −36, −86, 1, 1, −9, −75, 4, 14, 14, −16,2, 1, 33, 10},    {0, 4, −3, −55, 5, 9, 41, 30, 4, −14, 13, 85, −2, 10,−36, 39},    {1, −2, −6, 63, −3, 9, −9, −70, 7, −5, 49, 20, −4, −1, −32,58},    {1, 0, 11, −22, 0, 4, −15, 51, 3, −3, 23, −70, 3, 8, −2, 86}, },   {    {−86, 55, −9, 3, 66, −28, −8, 1, −11, −10, 11, −1, 3, 6, −1,−2},    {−58, −19, 26, −2, −28, 75, −30, 0, 46, −43, −10, 11, −7, −2,19, −5},    {45, −34, 29, −5, 69, −1, −34, 2, −7, −56, 32, 6, −26, 32,7, −11},    {−34, −72, 43, −1, 62, 16, 15, −18, −55, 43, −31, 7, 19, −4,−5, 9},    {19, −3, −36, 21, 50, 6, 36, −22, 30, −19, −83, 16, −7, −42,51, 6},    {30, 49, 11, −9, 2, 32, −59, 5, −52, 1, −14, 30, 52, −32, 27,−9},    {9, 18, 77, −44, 6, −42, −18, 5, 54, 33, −24, 11, −20, −29, 2,1},    {−21, −37, 6, 5, −32, −63, 7, −1, −3, −26, 34, 23, 37, −20, 61,−40},    {5, −26, −16, 4, 36, 13, −11, 24, 48, 12, 38, −36, 67, −51,−32, 5},    {14, 27, 43, 43, −1, 2, 29, −73, 24, −16, 7, 22, 49, 29,−31, −6},    {0, 12, 27, −49, −13, −6, 46, 11, −26, −66, −12, −40, 27,−13, 0, 55},    {9, 24, 18, −31, 19, 46, 59, 17, 5, 32, 26, −25, 8, 26,46, −54},    {1, 14, 33, 50, −7, 12, 3, −29, −25, 5, 46, −41, −50, −65,18, 12},    {−3, −2, −22, −59, 4, 16, 28, −32, −11, −12, 34, 61, −24,−47, −43, −22},    {−1, −4, −20, −31, 5, 6, −16, −44, 17, 33, 44, 14, 7,25, 50, 77},    {3, 7, 23, 41, 5, 10, 36, 76, 2, 3, 25, 69, −3, −2, 1,43},   }  },  { //2   {    {−112, 48, −1, 3, −28, 11, 1, 0, 19, −8, 0,0, 10, −4, 0, 0},    {−24, 8, −2, 1, 112, −42, −3, 0, 31, −11, −3, 1,−16, 6, 0, 0},    {37, 87, −73, 14, 10, 26, −20, 3, −11, −17, 14, −3,−6, −8, 6, −1},    {28, −6, −3, 1, −19, 9, −3, 1, 109, −38, −5, 0, 39,−13, −3, 1},    {9, 18, −14, 5, −32, −89, 65, −9, −10, −38, 27, −4, 4,10, −8, 2},    {−22, −58, −66, 78, −8, −16, −21, 23, 5, 16, 13, −17, 4,6, 6, −8},    {2, −2, 0, 0, −26, 6, 1, 1, 36, −8, 0, −1, −116, 28, 8,−1},    {11, 30, −16, −1, −10, −24, 15, −3, 28, 89, −51, 4, 13, 46, −25,−1},    {2, 8, 8, −12, −21, −51, −68, 61, −14, −30, −46, 40, 0, 3, 1,−3},    {16, 33, 63, 90, 7, 15, 26, 30, −5, −11, −25, −10, −4, −6, −13,−5},    {−1, −1, −16, 3, −4, −19, 24, −10, 6, 28, −40, 17, −23, −103,35, 5},    {−10, −24, −35, 9, 8, 28, 28, −30, −17, −44, −58, 46, −4, 11,−55, 30},    {−2, −6, −13, −17, 12, 26, 53, 83, 8, 18, 37, 60, 1, 0, −1,1},    {1, 4, 7, −5, −4, −13, −26, 16, 6, 20, 36, −24, −16, −43, −91,49},    {4, 10, 21, 37, −5, −12, −27, −46, 6, 19, 41, 77, 5, 10, 24,51},    {−1, −2, −4, −9, 2, 5, 13, 29, −3, −7, −21, −43, 7, 19, 47, 10}, },   {    {−98, 41, −1, 3, 66, −21, −4, 0, −14, −2, 5, −1, 1, 3, −1,0},    {59, 36, −29, 4, 36, −61, 21, −1, −61, 32, 3, −4, 22, −3, −8, 3},   {−8, 75, −46, 6, −56, −33, 24, 2, 58, −5, −2, −5, −18, 3, −1, 2},   {47, 4, 4, −4, 73, −11, −15, 4, 51, −38, 12, −2, −61, 22, 3, −2},   {−14, −50, −14, 20, 20, −21, 56, −26, 31, 55, −52, 8, −31, −23, 14,5},    {1, 14, 74, −51, −14, −49, −34, 30, 20, 50, −9, −3, −12, −18, 8,−1},    {18, 16, −3, 2, 38, 21, −10, 0, 61, 3, −15, 5, 89, −43, 10, −2},   {14, 53, −7, 2, 6, 61, −24, −3, −31, 19, −36, 18, −46, −47, 41, −7},   {−8, −22, −51, −18, 9, 22, −10, 52, 7, 37, 48, −54, −19, −41, −23,19},    {−3, −4, −22, −81, 8, 13, 47, 51, −11, −29, −47, 0, 11, 24, 19,−8},    {−8, −30, −16, 5, −13, −53, −12, 8, −13, −59, 14, 4, −4, −67,58, −18},    {5, 21, 50, −18, 2, 13, 65, −26, −8, −36, 16, −18, −11,−58, −42, 30},    {4, 7, 25, 49, −2, −3, −1, 25, −7, −16, −42, −85, 7,18, 34, 44},    {2, 11, 24, 0, 4, 20, −7, 6, 27, 65, −20, 7, 22, 72,−39},    {2, 5, 24, 59, 2, 3, 26, 77, −2, −4, −9, 20, −5, −12, −35,−61},    {0, 1, 7, 19, 1, 2, 16, 45, 2, 4, 27, 68, 2, 7, 27, 86},   } },  { //3   {    {114, −38, −3, −2, 20, 23, −12, 1, −22, 18, 4, −2, −5,−1, 5, 0},    {19, 43, −17, 2, −64, 59, 14, −6, −18, −34, 33, −1, 11,−20, −1, 7},    {−34, 29, −21, 2, 45, 55, −33, −7, −44, 58, 27, −17, −8,−21, 32, 0},    {31, 79, −42, 3, 56, −5, 31, −17, 42, −19, 7, 9, −10,20, −9, 6},    {13, 29, −31, 6, −49, −5, −39, 12, 25, 49, −61, −15, −12,41, 21, −33},    {11, 43, 20, −7, −7, −45, 46, 9, −85, 18, −31, 24, −22,−10, 2, −8},    {−10, −18, −78, 37, 12, −12, −37, 9, −48, −46, −21, 4,7, −3, −43, −10},    {18, 16, −15, 7, −8, −71, −21, −2, 6, −6, 27, −45,40, −55, 51, −1},    {8, 35, 43, −12, 26, 30, −27, 42, 4, −21, −46, 14,71, −26, −14, −21},    {8, 28, 40, 17, 1, −10, −56, 51, 5, −31, 20, −13,−77, −6, −6, 18},    {−5, −12, 6, −24, 17, 19, 16, −12, −11, −57, −7,−39, −27, 11, 43, −83},    {−3, −20, −24, 44, 2, 15, 29, 45, 28, 1, −12,62, −21, −47, 53, −13},    {1, 2, 17, 74, 4, 0, 33, 47, −12, 13, 38,−31, 33, 55, −3, −21},    {−3, −2, −19, −52, 0, −12, −22, 36, −20, −22,30, 46, 23, 61, 52, 23},    {2, 6, 33, 60, 5, 9, −20, −63, −14, −32,−35, 12, 7, 24, 50, 40},    {5, 11, 20, 16, −8, −20, −39, −42, 4, 13,49, 65, 4, 1, −19, −68},   }   {    {−96, 30, 5, 2, 67, −29, −4, −1,−18, 14, 1, −1, 5, −2, 0, 0},    {−15, −88, 29, 3, 36, 62, −38, −4, −18,−4, 24, 0, 3, −6, −5, 2},    {65, 24, −20, 0, 57, −3, 5, −2, −81, 16,14, 1, 26, −14, −6, 2},    {−29, 24, −39, 13, −17, 54, 35, −18, −24,−77, 19, 18, 20, 13, −29, −4},    {1, 41, 73, −34, −26, −6, −50, 36,−24, −39, 20, −15, 25, 13, −12, 0},    {−30, 38, 0, 9, −63, −28, 16, −3,−49, 27, −19, −1, 68, −29, −1, −2},    {11, −32, 22, −15, 31, −18, 31,16, −10, −41, −78, 13, 12, 54, 12, −30},    {12, −29, −25, 32, 8, −64,−33, −23, 19, −19, 40, −11, 29, 56, −37, 12},    {−10, −24, −57, −82, 3,−2, 14, 63, 6, 6, 21, −28, 5, 8, −14, 8},    {−7, 19, −43, 18, −8, 34,−51, 2, −6, 8, −23, 42, 27, 40, 72, 1},    {−13, −17, −14, 6, −33, −26,−14, 1, −70, −10, 5, −5, −90, 20, 9, −5},    {−3, 5, −13, −47, −11, 0,−29, −30, 0, 26, 22, 86, 10, 24, 19, −50},    {−3, −2, 37, −18, −5, 6,60, −24, −9, 22, 41, −1, 5, 54, 42, 59},    {5, −17, −10, −24, 11, −39,−7, −23, 8, −67, 14, 8, 5, −57, 68, 22},    {−2, 6, −10, −48, −3, 9,−34, −58, −7, 3, −57, 2, −6, 0, −40, 66},    {−1, 0, 12, 35, −1, 0, −18,68, −1, −2, −11, 74, 2, 4, 6, 66},   }  } };

All of the illustrative transform kernel matrices shown in Table 8 aretransform kernel matrices multiplied by 128 as a scaling value. In ag_aiNsst8×8[N1][N2][16][64] array present in matrix arrays of Table 8,N1 denotes the number of transform sets (N1 is 4 or 35, distinguished byindex 0, 1, . . . , and N1−1), N2 denotes the number (1 or 2) oftransform kernel matrices included in each transform set, and [16][64]denotes a 16×64 reduced secondary transform (RST).

As shown in Table 3 and Table 4, when a transform set includes onetransform kernel matrix, either a first transform kernel matrix or asecond transform kernel matrix may be used for the transform set inTable 8.

While 16 transform coefficients are output when the RST is applied, onlym transform coefficients may be output when only an m×64 portion of a16×64 matrix is applied. For example, when only eight transformcoefficients are output by setting m=8 and multiplying only an 8×64matrix from the top, it is possible to reduce computational amount byhalf. To reduce computational amount in a worst case, an 8×64 matrix maybe applied to an 8×8 transform unit (TU).

An m×64 transform matrix applicable to an 8×8 region (m≤16, e.g., thetransform kernel matrices in Table 8) receives 64 pieces of data andgenerates m coefficients. That is, as shown in Equation 5, when the 64pieces of data form a 64×1 vector, an m×1 vector is generated bysequentially multiplying an m×64 matrix and a 64×1 vector. Here, the 64pieces of data forming the 8×8 region may be properly arranged to form a64×1 vector. For example, as shown in Table 10, the data may be arrangedin the order of indexes indicated at respective positions in the 8×8region.

TABLE 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2425 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 4849 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

As shown in Table 10, the data is arranged in the row-first direction inthe 8×8 region for a secondary transform. This order refers to an orderin which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 10.

When there are 67 intra prediction modes as shown in FIG. 5, alldirectional modes (mode 2 to mode 66) are symmetrically configured aboutmode 34. That is, mode (2+n) is symmetric to mode (66−n) (0≤n≤31) aboutmode 34 in terms of prediction direction. Therefore, if a dataarrangement order for configuring a 64×1 input vector for mode (2+n),that is, modes 2 to 33, corresponds to the row-first direction as shownin Table 10, a 64×1 input vector for mode (66−n) may be configured in anorder shown in Table 11.

TABLE 11 1 9 17 25 33 41 49 57 2 10 18 26 34 42 50 58 3 11 19 27 35 4351 59 4 12 20 28 36 44 52 60 5 13 21 29 37 45 53 61 6 14 22 30 38 46 5462 7 15 23 31 39 47 55 63 8 16 24 32 40 48 56 64

As shown in Table 11, the data is arranged in the column-first directionin the 8×8 region for a secondary transform. This order refers to anorder in which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 11.

Table 11 shows that, for intra prediction mode (66−n), that is, formodes 35 to 66, a 64×1 input vector may be configured for according tothe column-first direction.

In summary, the same transform kernel matrix shown in Table 8 may beapplied while symmetrically arranging input data for mode (2+n)according to the row-first direction and input data for mode (66−n)(0≤n≤31) according to the column-first direction. A transform kernelmatrix to be applied in each mode is shown in Table 5 to Table 7. Here,either the arrangement order shown in Table 10 or the arrangement ordershown in Table 11 may be applied for the planar mode of intra predictionmode 0, the DC mode of intra prediction mode 1, and intra predictionmode 34. For example, for intra prediction mode 34, input data may bearranged according to the row-first direction as shown in Table 10.

According to another example, all of the illustrative transform kernelmatrices shown in Table 9 applicable to a 4×4 region are transformkernel matrices multiplied by 128 as a scaling value. In ag_aiNsst4×4[N1][N2][16][64] array present in matrix arrays of Table 9,N1 denotes the number of transform sets (N1 is 4 or 35, distinguished byindex 0, 1, . . . , and N1−1), N2 denotes the number (1 or 2) oftransform kernel matrices included in each transform set, and [16][16]denotes a 16×16 transform.

As shown in Table 3 and Table 4, when a transform set includes onetransform kernel matrix, either a first transform kernel matrix or asecond transform kernel matrix may be used for the transform set inTable 9.

As in the 8×8 RST, only m transform coefficients may be output when onlyan m×16 portion of a 16×16 matrix is applied. For example, when onlyeight transform coefficients are output by setting m=8 and multiplyingonly an 8×16 matrix from the top, it is possible to reduce computationalamount by half. To reduce computational amount in a worst case, an 8×16matrix may be applied to a 4×4 transform unit (TU).

Basically, the transform kernel matrices applicable to a 4×4 region,presented in Table 9, may be applied to a 4×4 TU, a 4×M TU, and an M×4TU (M>4, the 4×M TU and the M×4 TU may be divided into 4×4 regions, towhich each designated transform kernel matrix may be applied, or thetransform kernel matrices may be applied only to a maximum top-left 4×8or 8×4 region) or may be applied only to a top-left 4×4 region. If thesecondary transform is configured to be applied only to the top-left 4×4region, the transform kernel matrices applicable to an 8×8 region, shownin Table 8, may be unnecessary.

An m×16 transform matrix applicable to a 4×4 region (m≤16, e.g., thetransform kernel matrices in Table 9) receives 16 pieces of data andgenerates m coefficients. That is, when the 16 pieces of data form a16×1 vector, an m×1 vector is generated by sequentially multiplying anm×16 matrix and a 16×1 vector. Here, the 16 pieces of data forming the4×4 region may be properly arranged to form a 16×1 vector. For example,as shown in Table 12, the data may be arranged in the order of indexesindicated at respective positions in the 4×4 region.

TABLE 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

As shown in Table 12, the data is arranged in the row-first direction inthe 4×4 region for a secondary transform. This order refers to an orderin which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 12.

When there are 67 intra prediction modes as shown in FIG. 5, alldirectional modes (mode 2 to mode 66) are symmetrically configured aboutmode 34. That is, mode (2+n) is symmetric to mode (66−n) (0≤n≤31) aboutmode 34 in terms of prediction direction. Therefore, if a dataarrangement order for configuring a 16×1 input vector for mode (2+n),that is, modes 2 to 33, corresponds to the row-first direction as shownin Table 12, a 64×1 input vector for mode (66−n) may be configured in anorder shown in Table 13.

TABLE 13 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16

As shown in Table 13, the data is arranged in the column-first directionin the 4×4 region for a secondary transform. This order refers to anorder in which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 13.

Table 13 shows that, for intra prediction mode (66−n), that is, formodes 35 to 66, a 16×1 input vector may be configured for according tothe column-first direction.

In summary, the same transform kernel matrices shown in Table 9 may beapplied while symmetrically arranging input data for mode (2+n)according to the row-first direction and input data for mode (66−n)(0≤n≤31) according to the column-first direction. A transform kernelmatrix to be applied in each mode is shown in Table 5 to Table 7. Here,either the arrangement order shown in Table 12 or the arrangement ordershown in Table 13 may be applied for the planar mode of intra predictionmode 0, the DC mode of intra prediction mode 1, and intra predictionmode 34. For example, for intra prediction mode 34, input data may bearranged according to the row-first direction as shown in Table 12.

The following embodiments may be proposed in order to reducecomputational amount in a worst case. Hereinafter, a matrix including Mrows and N columns is expressed as an M×N matrix, and the M×N matrixrefers to a transform matrix applied in a forward transform, that is,when the encoding apparatus performs a transform (RST). Accordingly, inthe inverse transform (inverse RST) performed by the decoding apparatus,an N×M matrix obtained by transposing the M×N matrix may be used.

1) In a case of a block (e.g., a transform unit) having a width of W anda height of H where W≥8 and H≥8, a transform kernel matrix applicable toan 8×8 region is applied to a top-left 8×8 region of the block. In acase where W=8 and H=8, only an 8×64 portion of a 16×64 matrix may beapplied. That is, eight transform coefficients may be generated.

2) In a case of a block (e.g., a transform unit) having a width of W anda height of H where one of W and H is less than 8, that is, one of W andH is 4, a transform kernel matrix applicable to a 4×4 region is appliedto a top-left region of the block. In a case where W=4 and H=4, only an8×16 portion of a 16×16 matrix may be applied, in which case eighttransform coefficients are generated.

If (W, H)=(4, 8) or (8, 4), a secondary transform is applied only to thetop-left 4×4 region. If W or H is greater than 8, that is, if one of Wand H is equal to or greater than 16 and the other is 4, the secondarytransform is applied only to two top-left 4×4 blocks. That is, only atop-left 4×8 or 8×4 region may be divided into two 4×4 blocks, and adesignated transform kernel matrix may be applied thereto.

3) In a case of a block (e.g., a transform unit) having a width of W anda height of H where both W and H are 4, a secondary transform may not beapplied.

4) In a case of a block (e.g., a transform unit) having a width of W anda height of H, the number of coefficients generated by applying asecondary transform may be maintained to be ¼ or less of the area of thetransform unit (i.e., the total number of pixels included in thetransform unit=W×H). For example, when both W and H are 4, a top 4×16matrix of a 16×16 matrix may be applied so that four transformcoefficients are generated.

Assuming that a secondary transform is applied only to a top-left 8×8region of the entire transform unit (TU), eight or less coefficientsneed to be generated for a 4×8 transform unit or a 8×4 transform unit,and thus a top 8×16 matrix of a 16×16 matrix may be applied to a topleft 4×4 region. Up to a 16×64 matrix may be applied to an 8×8 transformunit (up to 16 coefficients can be generated). In a 4×N or N×4 (N≥16)transform unit, a 16×16 matrix may be applied to a top-left 4×4 block,or a top 8×16 matrix of the 16×16 matrix may be applied to two top-left4×4 blocks. Similarly, in a 4×8 transform unit or 8×4 transform unit,eight transform coefficients may be generated by applying a top 4×16matrix of the 16×16 matrix to two top-left 4×4 blocks.

5) The maximum size of a secondary transform applied to a 4×4 region maybe limited to 8×16. In this case, the amount of a memory required tostore transform kernel matrices applied to the 4×4 region can be reducedby half compared to that in a 16×16 matrix.

For example, in all transform kernel matrices shown in Table 9, themaximum size may be limited to 8×16 by extracting only a top 8×16 matrixof each 16×16 matrix, and an actual image coding system may beimplemented to store only 8×16 matrices of the transform kernelmatrices.

If the maximum applicable transform size is 8×16 and the maximum numberof multiplications required to generate one coefficient is limited to 8,an up to 8×16 matrix may be applied to a 4×4 block, and an up to 8×16matrix may be applied to each of up to two top-left two 4×4 blocksincluded in a 4×N block or an N×4 block (N≥8, N=2n, n≥3). For example,an 8×16 matrix may be stored for one top-left 4×4 block in a 4×N blockor an N×4 block (N≥8, N=2n, n≥3).

According to an embodiment, when coding an index specifying a secondarytransform to be applied to a luma component, specifically, when onetransform set includes two transform kernel matrices, it is necessary tospecify whether to apply the secondary transform and which transformkernel matrix to apply in the secondary transform. For example, when nosecondary transform is applied, a transform index may be coded as 0, andwhen the secondary transform is applied, transform indexes for twotransform sets may be coded as 1 and 2, respectively.

In this case, when coding the transform index, truncated unary codingmay be used. For example, binary codes of 0, 10, and 11 may berespectively allocated to transform indexes 0, 1, and 2, thereby codingthe transform indexes.

In addition, when coding the transform index by truncated unary coding,different CABAC context may be assigned to each bin. When coding thetransform indexes 0, 10, and 11 in the above example, two CABAC contextsmay be used.

When coding a transform index specifying a secondary transform to beapplied to a chroma component, specifically, when one transform setincludes two transform kernel matrices, it is necessary to specifywhether to apply the secondary transform and which transform kernelmatrix to apply in the secondary transform similarly to when coding thetransform index of the secondary transform for the luman component. Forexample, when no secondary transform is applied, a transform index maybe coded as 0, and when the secondary transform is applied, transformindexes for two transform sets may be coded as 1 and 2, respectively.

In this case, when coding the transform index, truncated unary codingmay be used. For example, binary codes of 0, 10, and 11 may berespectively allocated to transform indexes 0, 1, and 2, thereby codingthe transform indexes.

In addition, when coding the transform index by truncated unary coding,different CABAC context may be assigned to each bin. When coding thetransform indexes 0, 10, and 11 in the above example, two CABAC contextsmay be used.

According to an embodiment, a different CABAC context set may beallocated according to a chroma intra prediction mode. For example, whenchroma intra prediction modes are divided into non-directional modes,such as a planar mode or a DC mode, and other directional modes (i.e.,divided into two groups), a corresponding CABAC context set (includingtwo contexts) may be allocated for each group when coding 0, 10, and 11in the above example.

When the chroma intra prediction modes are divided into a plurality ofgroups and a corresponding CABAC context set is allocated, it isnecessary to find out a chroma intra prediction mode value before codingthe transform index of a secondary transform. However, in a chromadirect mode (DM), since a luma intra prediction mode value is used as itis, it is also necessary to find out an intra prediction mode value fora luma component. Therefore, when coding information on a chromacomponent, data dependency on luma component information may occur.Thus, in the chroma DM, when coding the transform index of the secondarytransform without having information on the intra prediction mode, thedata dependency can be removed by mapping to a specific group. Forexample, if the chroma intra prediction mode is the chroma DM, thetransform index may be coded using a corresponding CABAC context setassuming the planner mode or the DC mode, or a corresponding CABACcontext set may be applied assuming that other directional modes.

FIG. 9 is a flowchart illustrating an operation of a video decodingapparatus according to an embodiment of the present disclosure.

Each operation illustrated in FIG. 9 may be performed by the decodingapparatus 300 illustrated in FIG. 3. Specifically, S910 may be performedby the entropy decoder 310 illustrated in FIG. 3, S920 may be performedby the dequantizer 321 illustrated in FIGS. 3, S930 and S940 may beperformed by the inverse transformer 322 illustrated in FIG. 3, and S950may be performed by the adder 340 illustrated in FIG. 3. Operationsaccording to S910 to S950 are based on some of the foregoing detailsexplained with reference to FIG. 4 to FIG. 8. Therefore, a descriptionof specific details overlapping with those explained above withreference to FIG. 3 to FIG. 8 will be omitted or will be made briefly.

The decoding apparatus 300 according to an embodiment may derivequantized transform coefficients for a target block from a bitstream(S910). Specifically, the decoding apparatus 300 may decode informationon the quantized transform coefficients for the target block from thebitstream and may derive the quantized transform coefficients for thetarget block based on the information on the quantized transformcoefficients for the target block. The information on the quantizedtransform coefficients for the target block may be included in asequence parameter set (SPS) or a slice header and may include at leastone of information on whether a reduced transform (RST) is applied,information on a reduced factor, information on a minimum transform sizeto which the RST is applied, information on a maximum transform size towhich the RST is applied, information on a reduced inverse transformsize, and information on a transform index indicating any one oftransform kernel matrices included in a transform set.

The decoding apparatus 300 according to an embodiment may derivetransform coefficients by dequantizing the quantized transformcoefficients for the target block (S920).

The decoding apparatus 300 according to an embodiment may derivemodified transform coefficients based on an inverse reduced secondarytransform (RST) of the transform coefficients (S930).

In an example, the inverse RST may be performed based on an inverse RSTtransform matrix, and the inverse RST transform matrix may be anonsquare matrix in which the number of columns is less than the numberof rows.

In an embodiment, S930 may include decoding a transform index,determining whether a condition for applying an inverse RST is satisfiedbased on the transform index, selecting a transform kernel matrix, andapplying the inverse RST to the transform coefficients based on theselected transform kernel matrix and/or the reduced factor when thecondition for applying the inverse RST is satisfied. In this case, thesize of a reduced inverse transform matrix may be determined based onthe reduced factor.

The decoding apparatus 300 according to an embodiment may deriveresidual samples for the target block based on an inverse transform ofthe modified transform coefficients (S940).

The decoding apparatus 300 may perform an inverse primary transform onthe modified transform coefficients for the target block, in which casea reduced inverse transform may be applied or a conventional separabletransform may be used as the inverse primary transform.

The decoding apparatus 300 according to an embodiment may generatereconstructed samples based on the residual samples for the target blockand prediction samples for the target block (S950).

Referring to S930, it may be identified that the residual samples forthe target block are derived based on the inverse RST of the transformcoefficients for the target block. From the perspective of the size ofthe inverse transform matrix, since the size of a regular inversetransform matrix is N×N but the size of the inverse RST matrix isreduced to N×R, it is possible to reduce memory usage in a case ofperforming the inverse RST by an R/N ratio compared to that in a case ofperforming a regular transform. Further, using the inverse RST matrixcan reduce the number of multiplications (N×R) by the R/N ratio,compared to the number of multiplications N×N in a case of using theregular inverse transform matrix. In addition, since only R transformcoefficients need to be decoded when the inverse RST is applied, thetotal number of transform coefficients for the target block may bereduced from N to R, compared to that in a case where N transformcoefficients needs to be decoded when a regular inverse transform isapplied, thus increasing decoding efficiency. That is, according toS930, the (inverse) transform efficiency and decoding efficiency of thedecoding apparatus 300 may be increased through the inverse RST.

FIG. 10 is a control flowchart illustrating an inverse RST according toan embodiment of the present disclosure.

The decoding apparatus 300 receives information on quantized transformcoefficients, an intra prediction mode, and a transform index through abitstream (S1000).

Transform coefficients are derived from the quantized transformcoefficients received through the bitstream via dequantization as shownin S920 of FIG. 9.

To apply an inverse RST to the dequantized transform coefficients, atransform set and a transform kernel matrix to be applied to a targetblock are derived (S1010).

According to an example, the transform set may be derived based on amapping relationship according to an intra prediction mode for thetarget block, and a plurality of intra prediction modes may be mapped toone transform set. Each one transform set may include a plurality oftransform kernel matrices. A transform index may indicate any one of theplurality of transform kernel matrices. For example, when one transformset includes two transform kernel matrices, the transform index mayindicate any one of the two transform kernel matrices.

A syntax element of the transform index according to an embodiment mayindicate whether an inverse RST is applied and one of transform kernelmatrices included in the transform set. When the transform set includestwo transform kernel matrices, the syntax element of the transform indexmay have three values.

That is, according to an embodiment, the value of the syntax element ofthe transform index may include 0 indicating that the inverse RST is notapplied to the target block, 1 indicating a first transform kernelmatrix of the transform kernel matrices, and 2 indicating a secondtransform kernel matrix of the transform kernel matrices. Thisinformation is received as syntax information, and the syntaxinformation is received as a bin string including a 0 and a 1.

The transform kernel matrix according to an example may be applied to aspecified top-left region of the target block, for example, an 8×8region or a 4×4 region, according to the reduced or simplified size of asecondary transform, and the size of modified transform coefficientsoutput by applying the transform kernel matrix, that is, the number oftransform coefficients, may be derived based on the transform index, theintra prediction mode, and the size of the target block to which thesecondary transform is applied.

As shown in Equation 7, the transform coefficients in a one-dimensionalarray derived through the dequantization may be subjected to a matrixoperation with the transform kernel matrix, thereby deriving modifiedtransform coefficients in a two-dimensional array.

The inverse transformer 322 according to this embodiment may apply thetransform kernel matrix to the transform coefficients, and maytwo-dimensionally arrange the modified transform coefficients, which arevalues obtained by the operation with the transform coefficients and thetransform kernel matrix, in either the row-first direction or thecolumn-first direction according to the intra prediction mode (S1020).

That is, the inverse transformer 322 may generate the modified transformcoefficients of an 8×8 region or a 4×4 region in a two-dimensional blockby applying the inverse RST, and applies an inverse primary transform tothe modified transform coefficients of the generated two-dimensionalblock.

Here, when the intra prediction mode is a horizontal mode in a leftdirection based on a top-left diagonal direction, the modified transformcoefficients may be two-dimensionally arranged according to therow-first direction. That is, when the intra prediction mode applicableto the target block is any one of the 65 directional modes shown in FIG.5, an intra prediction mode in the top-left diagonal direction is mode34, and the intra prediction mode applied to the target block includesmode 2 to mode 34, the modified transform coefficients may betwo-dimensionally arranged according to the row-first direction.

However, when the intra prediction mode is a vertical mode in a rightdirection based on the top-left diagonal direction, the modifiedtransform coefficients may be two-dimensionally arranged according tothe column-first direction. That is, when the intra prediction modeapplied to the target block includes mode 35 to mode 66, the modifiedtransform coefficients may be two-dimensionally arranged according tothe column-first direction

When the intra prediction mode applied to the target block is anon-directional mode, such as the planar mode or the DC mode, themodified transform coefficients may be two-dimensionally arranged ineither the row-first direction or the column-first direction. Accordingto an example, the modified transform coefficients may betwo-dimensionally arranged according to the row-first direction.

FIG. 11 is a flowchart illustrating an operation of a video encodingapparatus according to an embodiment of the present disclosure.

Each operation illustrated in FIG. 11 may be performed by the encodingapparatus 200 illustrated in FIG. 2. Specifically, S1110 may beperformed by the predictor 220 illustrated in FIG. 2, S1120 may beperformed by the subtractor 231 illustrated in FIGS. 2, S1130 and S1140may be performed by the transformer 232 illustrated in FIG. 2, and S1150may be performed by the quantizer 233 and the entropy encoder 240illustrated in FIG. 2. Operations according to S1110 to S1150 are basedon some of contents described in FIG. 4 to FIG. 8. Therefore, adescription of specific details overlapping with those explained abovewith reference to FIG. 4 to FIG. 8 will be omitted or will be madebriefly.

The encoding apparatus 200 according to an embodiment may deriveprediction samples based on an intra prediction mode applied to a targetblock (S1110).

The encoding apparatus 200 according to an embodiment may deriveresidual samples for the target block (S1120).

The encoding apparatus 200 according to an embodiment may derivetransform coefficients for the target block based on primary transformof the residual sample (S1130). The primary transform may be performedthrough a plurality of transform kernels, and the transform kernels maybe selected based on the intra prediction mode.

The decoding apparatus 300 may perform a secondary transform,specifically an NSST, on the transform coefficients for the targetblock, in which case the NSST may be performed based on a reducedtransform (RST) or without being based on the RST. When the NSST isperformed based on the reduced transform, an operation according toS1140 may be performed.

The encoding apparatus 200 according to an embodiment may derivemodified transform coefficients for the target block based on the RST ofthe transform coefficients (S1140). In an example, the RST may beperformed based on a reduced transform matrix or a transform kernelmatrix, and the reduced transform matrix may be a nonsquare matrix inwhich the number of rows is less than the number of columns.

In an embodiment, S1140 may include determining whether a condition forapplying the RST is satisfied, generating and encoding the transformindex based on the determination, selecting a transform kernel, andapplying the RST to the residual samples based on the selected transformkernel matrix and/or a reduced factor when the condition for applyingthe RST is satisfied. In this case, the size of the reduced transformkernel matrix may be determined based on the reduced factor.

The encoding apparatus 200 according to an embodiment may derivequantized transform coefficients by performing quantization based on themodified transform coefficients for the target block and may encodeinformation on the quantized transform coefficients (S1150).

Specifically, the encoding apparatus 200 may generate the information onthe quantized transform coefficients and may encode the generatedinformation on the quantized transform coefficients.

In an example, the information on the quantized transform coefficientsmay include at least one of information on whether the RST is applied,information on the reduced factor, information on a minimum transformsize to which the RST is applied, and information on a maximum transformsize to which the RST is applied.

Referring to S1140, it may be identified that the transform coefficientsfor the target block are derived based on the RST of the residualsamples. From the perspective of the size of the transform kernelmatrix, since the size of a regular transform kernel matrix is N×N butthe size of the reduced transform matrix is reduced to R×N, it ispossible to reduce memory usage in a case of performing the RST by anR/N ratio compared to that in a case of performing a regular transform.Further, using the reduced transform kernel matrix can reduce the numberof multiplications (R×N) by the R/N ratio, compared to the number ofmultiplications N×N in a case of using the regular transform kernelmatrix. In addition, since only R transform coefficients are derivedwhen the RST is applied, the total number of transform coefficients forthe target block may be reduced from N to R, compared to that in a casewhere N transform coefficients are derived when a regular transform isapplied, thus reducing the amount of data transmitted by the encodingapparatus 200 to the decoding apparatus 300. That is, according toS1140, the transform efficiency and coding efficiency of the encodingapparatus 320 may be increased through the RST.

FIG. 12 is a control flowchart illustrating an RST according to anembodiment of the present disclosure.

First, the encoding apparatus 200 may determine a transform set based ona mapping relationship according to an intra prediction mode applied toa target block (S1200).

The transformer 232 may select any one of a plurality of transformkernel matrices included in the transform set (S1210).

According to an example, the transform set may be derived based on themapping relationship according to the intra prediction mode of thetarget block, and a plurality of intra prediction modes may be mapped toone transform set. Each one transform set may include a plurality oftransform kernel matrices. When one transform set includes two transformkernel matrices, a transform index indicating any one of the twotransform kernel matrices may be encoded and may be signaled to thedecoding apparatus.

When two transforms are applied to a residual sample, the residualsample may be referred to as a transform coefficient after beingsubjected to a primary transform, and may be referred to as a modifiedtransform coefficient after being subjected to the primary transform andthen a secondary transform, such as an RST.

When performing the RST on transform coefficients using the transformkernel matrix, the transformer 232 one-dimensionally arrange thetransform coefficients in a two-dimensional array, which have beensubjected to the primary transform, according to either the row-firstdirection or the column-first direction, based on the intra predictionmode applied to the target block (1220).

The transform kernel matrix may be applied to a specified top-leftregion of the target block, for example, an 8×8 region or a 4×4 region,according to the reduced or simplified size of a secondary transform,and the size of modified transform coefficients output by applying thetransform kernel matrix, that is, the number of modified transformcoefficients, may be derived based on the size of the transform kernelmatrix, the intra prediction mode, and the size of the target block towhich the secondary transform is applied.

The two-dimensional transform coefficients need to be one-dimensionallyarranged for a matrix operation with the transform kernel matrix asshown in Equation 5, and a smaller number of modified transformcoefficients than that of transform coefficients may be derived throughan operation, such as Equation 6.

That is, the transform coefficients in the two-dimensional array in thespecified region may be read in one dimension according to a certaindirection, from which modified transform coefficients are derivedthrough the matrix operation with the transform kernel matrix.

When the RST is performed, information on the RST may be encoded by theentropy encoder 240.

First, the entropy encoder 240 may derive a syntax element value for thetransform index indicating any one of the transform kernel matricesincluded in the transform set, may binarize the derived syntax elementvalue for the transform index, and may encode bins of a syntax elementbin string based on context information, that is, a context model, on abin string of the transform index.

The encoded bin string of the syntax element may be output as abitstream to the decoding apparatus 300 or to the outside.

In the above-described embodiments, the methods are explained on thebasis of flowcharts by means of a series of steps or blocks, but thepresent disclosure is not limited to the order of steps, and a certainstep may be performed in order or step different from that describedabove, or concurrently with another step. Further, it may be understoodby a person having ordinary skill in the art that the steps shown in aflowchart are not exclusive, and that another step may be incorporatedor one or more steps of the flowchart may be removed without affectingthe scope of the present disclosure.

The above-described methods according to the present disclosure may beimplemented as a software form, and an encoding apparatus and/ordecoding apparatus according to the disclosure may be included in adevice for image processing, such as, a TV, a computer, a smartphone, aset-top box, a display device or the like.

When embodiments in the present disclosure are embodied by software, theabove-described methods may be embodied as modules (processes, functionsor the like) to perform the above-described functions. The modules maybe stored in a memory and may be executed by a processor. The memory maybe inside or outside the processor and may be connected to the processorin various well-known manners. The processor may include anapplication-specific integrated circuit (ASIC), other chipset, logiccircuit, and/or a data processing device. The memory may include aread-only memory (ROM), a random access memory (RAM), a flash memory, amemory card, a storage medium, and/or other storage device. That is,embodiments described in the present disclosure may be embodied andperformed on a processor, a microprocessor, a controller or a chip. Forexample, function units shown in each drawing may be embodied andperformed on a computer, a processor, a microprocessor, a controller ora chip.

Further, the decoding apparatus and the encoding apparatus to which thepresent disclosure is applied, may be included in a multimediabroadcasting transceiver, a mobile communication terminal, a home cinemavideo device, a digital cinema video device, a surveillance camera, avideo chat device, a real time communication device such as videocommunication, a mobile streaming device, a storage medium, a camcorder,a video on demand (VoD) service providing device, an over the top (OTT)video device, an Internet streaming service providing device, athree-dimensional (3D) video device, a video telephony video device, anda medical video device, and may be used to process a video signal or adata signal. For example, the over the top (OTT) video device mayinclude a game console, a Blu-ray player, an Internet access TV, a Hometheater system, a smartphone, a Tablet PC, a digital video recorder(DVR) and the like.

In addition, the processing method to which the present disclosure isapplied, may be produced in the form of a program executed by acomputer, and be stored in a computer-readable recording medium.Multimedia data having a data structure according to the presentdisclosure may also be stored in a computer-readable recording medium.The computer-readable recording medium includes all kinds of storagedevices and distributed storage devices in which computer-readable dataare stored. The computer-readable recording medium may include, forexample, a Blu-ray Disc (BD), a universal serial bus (USB), a ROM, aPROM, an EPROM, an EEPROM, a RAM, a CD-ROM, a magnetic tape, a floppydisk, and an optical data storage device. Further, the computer-readablerecording medium includes media embodied in the form of a carrier wave(for example, transmission over the Internet). In addition, a bitstreamgenerated by the encoding method may be stored in a computer-readablerecording medium or transmitted through a wired or wirelesscommunication network. Additionally, the embodiments of the presentdisclosure may be embodied as a computer program product by programcodes, and the program codes may be executed on a computer by theembodiments of the present disclosure. The program codes may be storedon a computer-readable carrier.

FIG. 13 illustrates the structure of a content streaming system to whichthe present disclosure is applied.

Further, the contents streaming system to which the present disclosureis applied may largely include an encoding server, a streaming server, aweb server, a media storage, a user equipment, and a multimedia inputdevice.

The encoding server functions to compress to digital data the contentsinput from the multimedia input devices, such as the smart phone, thecamera, the camcoder and the like, to generate a bitstream, and totransmit it to the streaming server. As another example, in a case wherethe multimedia input device, such as, the smart phone, the camera, thecamcoder or the like, directly generates a bitstream, the encodingserver may be omitted. The bitstream may be generated by an encodingmethod or a bitstream generation method to which the present disclosureis applied. And the streaming server may store the bitstream temporarilyduring a process to transmit or receive the bitstream.

The streaming server transmits multimedia data to the user equipment onthe basis of a user's request through the web server, which functions asan instrument that informs a user of what service there is. When theuser requests a service which the user wants, the web server transfersthe request to the streaming server, and the streaming server transmitsmultimedia data to the user. In this regard, the contents streamingsystem may include a separate control server, and in this case, thecontrol server functions to control commands/responses betweenrespective equipments in the content streaming system.

The streaming server may receive contents from the media storage and/orthe encoding server. For example, in a case the contents are receivedfrom the encoding server, the contents may be received in real time. Inthis case, the streaming server may store the bitstream for apredetermined period of time to provide the streaming service smoothly.

For example, the user equipment may include a mobile phone, a smartphone, a laptop computer, a digital broadcasting terminal, a personaldigital assistant (PDA), a portable multimedia player (PMP), anavigation, a slate PC, a tablet PC, an ultrabook, a wearable device(e.g., a watch-type terminal (smart watch), a glass-type terminal (smartglass), a head mounted display (HMD)), a digital TV, a desktop computer,a digital signage or the like. Each of servers in the contents streamingsystem may be operated as a distributed server, and in this case, datareceived by each server may be processed in distributed manner.

What is claimed is:
 1. An image decoding method performed by a decodingapparatus, the method comprising: deriving transform coefficientsthrough dequantization based on quantized transform coefficients for atarget block; deriving modified transform coefficients based on aninverse reduced secondary transform (RST) using a preset transformkernel matrix for the transform coefficients; deriving residual samplesfor the target block based on an inverse primary transform for themodified transform coefficients; and generating a reconstructed picturebased on the residual samples for the target block, wherein the inverseRST is an inverse transform of a transform that derives a smaller numberof transform coefficients than transform coefficients for apredetermined top-left area of the target block based on the transformcoefficients for the top-left area, wherein the modified transformcoefficients in a two-dimensional array are derived, by applying theinverse RST, from the transform coefficients in a one-dimensional (1D)array according to one of a row-first direction or a column-firstdirection according to an intra prediction mode applied to the targetblock.
 2. The image decoding method of claim 1, wherein, when the intraprediction mode is a horizontal mode in a left direction with respect toa top-left diagonal direction, the modified transform coefficients aretwo-dimensionally arranged according to the row-first direction.
 3. Theimage decoding method of claim 2, wherein, when an intra prediction modeapplicable to the target block is one of 65 directional modes, an intraprediction mode in the top-left diagonal direction is mode 34, and theintra prediction mode applied to the target block comprises mode 2 tomode 34, the modified transform coefficients are two-dimensionallyarranged according to the row-first direction.
 4. The image decodingmethod of claim 1, wherein, when the intra prediction mode is a verticalmode in a right direction with respect to a top-left diagonal direction,the modified transform coefficients are two-dimensionally arrangedaccording to the column-first direction.
 5. The image decoding method ofclaim 3, wherein, when an intra prediction mode applicable to the targetblock is one of 65 directional modes, an intra prediction mode in thetop-left diagonal direction is mode 34, and the intra prediction modeapplied to the target block comprises mode 35 to mode 66, the modifiedtransform coefficients are two-dimensionally arranged according to thecolumn-first direction.
 6. The image decoding method of claim 1,wherein, when the intra prediction mode applied to the target block is aplanar mode or a DC mode, the modified transform coefficients aretwo-dimensionally arranged according to the row-first direction.
 7. Theimage decoding method of claim 1, wherein the inverse RST is performedbased on a transform set determined based on a mapping relationshipaccording to the intra prediction mode applied to the target block and atransform kernel matrix selected from among a plurality of transformkernel matrices comprised in each of the transform set, and is performedbased on a transform index indicating whether the inverse RST is appliedand one of the transform kernel matrices comprised in the transform set.8. An image encoding method performed by an image encoding apparatus,the method comprising: deriving prediction samples based on an intraprediction mode applied to a target block; deriving residual samples forthe target block based on the prediction samples; deriving transformcoefficients for the target block based on a primary transform for theresidual samples; deriving modified transform coefficients based on areduced secondary transform (RST) for the transform coefficients; andderiving quantized transform coefficients by performing quantizationbased on the modified transform coefficients, wherein the RST is atransform that derives a smaller number of transform coefficients thantransform coefficients for a predetermined top-left area of the targetblock based on the transform coefficients for the top-left area, whereinthe deriving of the modified transform coefficients one-dimensionallycomprises arranging the transform coefficients according to one of arow-first direction or a column-first direction according to the intraprediction mode applied to the target block and applying the RST to thetransform coefficients.
 9. The image encoding method of claim 8,wherein, when the intra prediction mode is a horizontal mode in a leftdirection with respect to a top-left diagonal direction, the transformcoefficients are one-dimensionally arranged according to the row-firstdirection.
 10. The image encoding method of claim 9, wherein, when anintra prediction mode applicable to the target block is one of 65directional modes, an intra prediction mode in the top-left diagonaldirection is mode 34, and the intra prediction mode applied to thetarget block comprises mode 2 to mode 34, the transform coefficients areone-dimensionally arranged according to the row-first direction.
 11. Theimage encoding method of claim 8, wherein, when the intra predictionmode is a vertical mode in a right direction with respect to a top-leftdiagonal direction, the transform coefficients are one-dimensionallyarranged according to the column-first direction.
 12. The image encodingmethod of claim 11, wherein, when an intra prediction mode applicable tothe target block is one of 65 directional modes, an intra predictionmode in the top-left diagonal direction is mode 34, and the intraprediction mode applied to the target block comprises mode 35 to mode66, the transform coefficients are one-dimensionally arranged accordingto the column-first direction.
 13. The image encoding method of claim 8,wherein, when the intra prediction mode applied to the target block is aplanar mode or a DC mode, the 8 transform coefficients areone-dimensionally arranged according to the row-first direction.
 14. Anon-transitory computer readable storage medium storing encodedinformation causing a decoding apparatus to perform an image decodingmethod, the method comprising: deriving transform coefficients throughdequantization based on quantized transform coefficients for a targetblock; deriving modified transform coefficients based on an inversereduced secondary transform (RST) using a preset transform kernel matrixfor the transform coefficients; deriving residual samples for the targetblock based on an inverse primary transform for the modified transformcoefficients; and generating a reconstructed picture based on theresidual samples for the target block, wherein the inverse RST is aninverse transform of a transform that derives a smaller number oftransform coefficients than transform coefficients for a predeterminedtop-left area of the target block based on the transform coefficientsfor the top-left area, wherein the modified transform coefficients in atwo-dimensional array are derived, by applying the inverse RST, from thetransform coefficients in a one-dimensional (1D) array according to oneof a row-first direction or a column-first direction according to anintra prediction mode applied to the target block.